Graduate School Form 30
Updated 1/15/2015
PURDUE UNIVERSITY
GRADUATE SCHOOL
Thesis/Dissertation Acceptance
This is to certify that the thesis/dissertation prepared
By Taylor Hines
Entitled
THE RADIUS OF COMPARISON AND MEAN DIMENSION
For the degree of Doctor of Philosophy
Is approved by the final examining committee:
Andrew Toms
Chair
Marius Dadarlat
D. Ben McReynolds
Lawrence Brown
To the best of my knowledge and as understood by the student in the Thesis/Dissertation
Agreement, Publication Delay, and Certification Disclaimer (Graduate School Form 32),
this thesis/dissertation adheres to the provisions of Purdue University’s “Policy of
Integrity in Research” and the use of copyright material.
Approved by Major Professor(s): Andrew Toms
Approved by: David Goldberg
Head of the Departmental Graduate Program
7/21/2015
Date
THE RADIUS OF COMPARISON AND MEAN DIMENSION
A Dissertation
Submitted to the Faculty
of
Purdue University
by
Taylor Hines
In Partial Fulfillment of the
Requirements for the Degree
of
Doctor of Philosophy
August 2015
Purdue University
West Lafayette, Indiana
ii
To my partner Susie, whose love and support made this dissertation possible.
Without you I am lost.
iii
ACKNOWLEDGMENTS
This work could not have progressed without the help and guidance of my advisor
Andrew Toms. I am grateful for all of the time, advice, and support he has given
me.
iv
TABLE OF CONTENTS
Page
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . .
1.1 Topological Dynamical Systems and Mean Dimension . . . .
1.2 Crossed Product C ∗ -algebras and the Radius of Comparison
1.3 Connections Between the Two and Summary of Results . . .
1
2
4
5
CHAPTER 2. TOPOLOGICAL DYNAMICAL SYSTEMS
DIMENSION . . . . . . . . . . . . . . . . . . . . . . .
2.1 Mean Dimension . . . . . . . . . . . . . . . . . .
2.2 Orbit Capacity and the Small Boundary Property
2.3 Flaws in the Mean Dimension . . . . . . . . . . .
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AND THE MEAN
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CHAPTER 3. CROSSED PRODUCTS AND THE RADIUS OF COMPARISON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 The Radius of Comparison . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Radius of Comparison from a Geometric Perspective . . . . . .
3.3 The Radius of Comparison of a Crossed Product . . . . . . . . . . .
3.4 Classification of Crossed Products . . . . . . . . . . . . . . . . . . .
3.4.1 Recursive subhomogeneous algebras . . . . . . . . . . . . . .
3.4.2 Large Subalgebras . . . . . . . . . . . . . . . . . . . . . . .
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CHAPTER 4. EXAMPLES OF
4.1 Inverse Limit Systems
4.2 Skew Product Systems
4.3 Giol-Kerr Systems . .
4.4 Disjoint systems . . . .
4.5 Connected systems . .
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CHAPTER 5. MEAN DIMENSION AND Z-STABILITY . . . . . . . . . .
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CHAPTER 6. MEAN DIMENSION AS AN UPPER BOUND ON THE RADIUS OF COMPARISON . . . . . . . . . . . . . . . . . . . . . . . . . .
53
CHAPTER 7. MEAN DIMENSION AS A LOWER BOUND ON THE RADIUS OF COMPARISON . . . . . . . . . . . . . . . . . . . . . . . . . .
62
MINIMAL DYNAMICAL SYSTEMS
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72
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
APPENDIX A. C ∗ -ALGEBRAS AND K-THEORY . . . . .
A.1 C ∗ -algebras . . . . . . . . . . . . . . . . . . . . . .
A.2 K-Theory . . . . . . . . . . . . . . . . . . . . . . .
A.2.1 The K0 -group of a unital C ∗ -algebra . . . .
A.2.2 The K0 -group from an A-module perspective
A.2.3 Topological K-theory . . . . . . . . . . . . .
A.2.4 Topological K-theory and cohomology . . .
A.2.5 The K1 -group . . . . . . . . . . . . . . . . .
A.3 The Classification Program . . . . . . . . . . . . .
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APPENDIX B. TOPOLOGICAL DYNAMICAL SYSTEMS
B.1 Decomposition of Dynamical Systems . . . . . . . .
B.2 Cantor Minimal Systems . . . . . . . . . . . . . . .
B.3 Disjoint Systems . . . . . . . . . . . . . . . . . . .
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APPENDIX C. CROSSED PRODUCT C ∗ -ALGEBRAS . . . . . . . . . . .
97
APPENDIX D. THE CUNTZ SEMIGROUP . . . . . . . . . . . . . . . . . .
D.1 The Cuntz Semigroup from an A-module perspective . . . . . . . .
100
101
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
7.1
7.2
7.3
Obstructions in the Cuntz semigroup of
Structure of Giol-Kerr crossed products
The proof of Theorem 1.3 . . . . . . .
7.3.1 Further Results . . . . . . . . .
a commutative C
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∗
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vi
LIST OF FIGURES
Figure
Page
3.1
Dynamic comparison in Lemma 3.4. . . . . . . . . . . . . . . . . . . . .
19
6.1
Illustration of Lemma 6.1. . . . . . . . . . . . . . . . . . . . . . . . . .
54
7.1
Commutative diagram used in the proof of Theorem 1.3. . . . . . . . .
63
7.2
Illustration of Theorem 1.3. . . . . . . . . . . . . . . . . . . . . . . . .
71
vii
SYMBOLS
∼M vN
Murray-von Neumann equivalence of projections (cf. Definition
A.5)
∼Cu
Cuntz equivalence of positive elements, also written ∼ (cf. Definition D.1)
.Cu
Cuntz subequivalence of positive elements, also written . (cf.
Definition D.1)
.B
Cuntz subequivalence with respect to a subalgebra B (cf. Example D.1.4)
[a]Cu
Cuntz class in Cu(A) of a positive element a (also denoted [a], cf.
Definition D.1)
A+
Subset of positive operators of a C ∗ -algebra A
A ⊕C B
Pullback C ∗ -algebra (cf. Definition 3.3)
AK
Putnam subalgebra (cf. Definition 3.4)
B
Bounded operators on Hilbert space
BA (E, F )
Adjointable linear maps between A-modules E and F
C
The complex numbers
ch
Chern character (cf. Section A.2.4)
C(X)
Continuous complex-valued functions on a compact space X
C0 (X)
Continuous complex-valued functions on a locally compact space
X vanishing at infinity
C(X) ⋊α Z
Crossed product C ∗ -algebra corresponding to the topological dynamical system (X, α) (cf. Definition C.2)
Cu(A)
Cuntz semigroup of a C ∗ -algebra A (cf. Definition D.1)
D(U)
Dimension of an open cover U (cf. Definition 2.2)
dim(X)
Covering dimension of a topological space X (cf. Definition 2.1)
viii
dτ
State on the Cuntz semigroup induced by a quasitrace τ (cf. Definition 3.1)
ei,j
The i, j-th matrix unit in Mn
Homeo(X)
Group of homeomorphisms of a topological space X
K
Compact operators on Hilbert space
K0 (A)
K0 -group of a C ∗ -algebra A (cf. Definition A.8)
K0 (A)+
Positive cone of the K0 -group of a C ∗ -algebra A
KA (E, F )
Compact linear maps between A-modules E and F
(K Z , σ)
Full shift on a finite simplicial complex K (cf. Definition B.1)
K(U)
Nerve of an open cover U (also denoted nerve(U), cf. Definition
2.1)
mdim(X, α)
Mean dimension of a dynamical system (X, α) (cf. Definition 2.3)
Mn
n × n matrices with complex entries
Mn (A)
n × n matrices with entries in a C ∗ -algebra A
M∞ (A)
Matrices of arbitrary size with entries in a C ∗ -algebra A
ocap(E)
Orbit capacity of a subset E of a topological dynamical system
(cf. Definition 2.4)
ord(U)
Order of an open cover U (cf. Definition 2.2)
[p]0
Class in K0 (A) of a projection p ∈ M∞ (A)
p×n
n-fold Cartesian product of a vector bundle (cf. Example A.9.7)
P∞ (A)
Set of projections in M∞ (A) (cf. Definition A.5)
Prob(X)
Probability measures on a space X
QT2 (A)
2-quasitraces on a C ∗ -algebra A (also denoted QT(A), cf. Definition A.4)
rc(A)
Radius of comparison of a C ∗ -algebra A (cf. Definition 3.1)
rank
Rank (or rank function) of an operator
Sn
n-sphere (S 1 is also denoted T)
σ(a)
Spectrum of an element a (cf. Definition A.3)
supp
Support of a continuous funtion (defined as the open set {f > 0})
ix
tn
Trivial vector bundle of rank n (cf. Example A.9.1)
tr
Trace of a matrix
V≺U
Open cover V refines the open cover U
V (A)
Semigroup of Murray-von Neumann equivalence classes of projections in M∞ (A) (cf. Definition A.6)
Vect(X)
Vector bundles over a topological space X
Widim
Width dimension (cf. Definition 2.2)
(X × Y, α ⋉ f )
A skew product dynamical system (cf. Definition 4.1)
Z
Jiang-Su algebra (cf. Theorem A.9)
x
ABSTRACT
Hines, Taylor PhD, Purdue University, August 2015. The Radius of Comparison and
Mean Dimension. Major Professor: Andrew Toms.
This dissertation is a collection of results and examples designed to support a single
conjecture, namely, that two dimensional invariants (the mean dimension for topological dynamical systems and the radius of comparison for C ∗ -algebras) are related by a
simple equation. Specifically, we conjecture that mdim(X, α) = 2 rc(C(X)⋊α Z) when
the dynamical system (X, α) is minimal. With three main Theorems and many examples and supporting results, we verify that this Conjecture is true in many cases. In the
case where (X, α) is not minimal, we show with several important examples that the
radius of comparison provides a more nuanced measurement than the mean dimension.
This naturally leads to the introduction of a new dimension theory for topological
dynamical systems: the dynamic dimension, defined by ddim(X, α) = rc(C(X)⋊α Z),
which extends the mean dimension in the minimal case and improves it in the nonminimal case. Furthermore, our results give an essential tool for computing the
radius of comparison of certain crossed product C ∗ -algebras, with important consequences for the Elliott classification program. In particular, our results show that
there exist infinite-dimensional dynamical systems with crossed products satisfying
the Toms-Winter conjecture, and are hence amenable to classification by K-theoretic
invariants.
1
CHAPTER 1. Introduction
The objective of this dissertation is to prove a relationship between two dimensional
invariants: the mean dimension of a topological dynamical system and the radius
of comparison of a C ∗ -algebra. Using the crossed product construction as a bridge
between topological dynamics and C ∗ -algebras, we show that the mean dimension
(a topological invariant) and the radius of comparison (an algebraic invariant) are
the same in many cases. The mean dimension is a purely topological invariant which
measures both the size and complexity of a topological dynamical system. The radius
of comparison is a purely algebraic invariant which was developed as part of Elliott’s
classification program to measure the complexity of a C ∗ -algebra. The crossed product functor maps a dynamical system (X, α) to an associated C ∗ -algebra C(X) ⋊α Z.
Our overarching conjecture is that the radius of comparison (on the algebraic side)
and the mean dimension (on the topological side) are related by the following equation:
Conjecture 1. If (X, α) is a minimal topological dynamical system, then
mdim(X, α) = 2 rc (C(X) ⋊α Z) .
This conjecture would allow for the creation of a new dimensional invariant, the
dynamic dimension, a universal dimension theory that would extend the definition of
mean dimension (while simultaneously addressing its flaws) and establish a powerful
bond between the fields of dynamical systems and C ∗ -algebras. Although Conjecture
1 is not proved in full generality, this dissertation makes significant progress towards
its resolution.
This dissertation is organized as follows: in the remainder of Chapter 1 we give
brief introductions to the mean dimension and the radius of comparison, explain the
2
connections we hope to illuminate, and describe the implications our results have for
topological dynamics, C ∗ -algebras, and the classification program.
In Chapter 2, we give a full introduction to the mean dimension, the associated
notion of orbit capacity, and describe the flaws in the mean dimension that we hope
to address with our main results.
In Chapter 3, we give a complete description of the radius of comparison for
crossed product algebras (as well as its geometric interpretation) and collect many of
the C ∗ -algebraic results we will need to prove our main results.
In Chapter 4, we give examples to show that the main results do indeed advance
the current results from the classification program for crossed products. Chapter 5
contains our first major result (Theorem 1.1), showing that zero mean dimension and
zero radius of comparison are equivalent when the property of Z-stability holds.
In Chapter 6 we prove Theorem 1.2, showing that the radius of comparison is
bounded above by the mean dimension in many cases. In the proof of Theorem 1.2
we also explain the use of Cantor minimal systems (the topological analogue of UHF
algebras) as spacing devices in the corresponding crossed products.
Finally, in Chapter 7 we prove that the radius of comparison is bounded below
by the mean dimension in the case of Giol-Kerr dynamical systems. As a result, we
show that the radius of comparison and the mean dimension are equal in this case.
Background material on C ∗ -algebras, topological dynamics, the crossed product
construction, and the Cuntz semigroup is included in the Appendices A-D (resp.)
1.1
Topological Dynamical Systems and Mean Dimension
By a dynamical system we mean a pair (X, α), where X is a compact metriz-
able space and α is a homeomorphism of X (see Appendix B for a more complete
introduction). Important examples include:
1. Rotation on the circle T ⊂ C, where α : T → T is defined by α(z) = e2πiθ z for
fixed rotation angle 0 ≤ θ < 1.
3
2. The full shift on a product space X = Y Z , where α : : X → X is defined by
α(yn ) = (yn+1 ).
3. Any continuous Z-action on a space X is equivalent to a topological dynamical
system, with n · x = αn (x). Topological flows defined by more general group
actions can also be considered, but this is a secondary objective of our project.
Of special importance are minimal dynamical systems, in which the orbit of every
point is dense. A common example is the rotation map on the circle, which is minimal
when the rotation angle θ is irrational. Minimal systems do not admit any nontrivial
closed subsystems and hence can be thought of as building blocks for other systems.
The mean dimension of a dynamical system (X, α), denoted mdim(X, α), is a
dimension theory first introduced by Gromov [21, Section 1.5] and later explored
by Lindenstrauss and Weiss [34]. Developed as an ‘equivariant covering dimension,’
the mean dimension captures both the size of the underlying space as well as the
complexity of the dynamics. For example:
1. If α is the full shift on X = Y Z , then mdim(X, α) = dim(Y ).
2. For any n ≥ 1, mdim(X, αn ) = n mdim(X, α).
3. If dim(X) is finite, then mdim(X, α) = 0 for any α.
4. If (X, α) has finite topological entropy, then mdim(X, α) = 0.
The concept of mean dimension can be defined for any group action, and has been
used to greatest effect in Lindenstrauss and Weiss’ work on infinite-entropy dynamical
systems. However, the concept of mean dimension remains limited, since it provides
no information about systems with finite entropy or any finite-dimensional systems.
Furthermore, the mean dimension degenerates in the presence of periodic points (as
discussed in [33]). These are issues our work will address.
4
1.2
Crossed Product C ∗ -algebras and the Radius of Comparison
A C ∗ -algebra is a Banach algebra over C with an involution a 7→ a∗ satisfying
the identity ka∗ ak = kak2 (see Appendix A for a brief introduction to C ∗ -algebras).
Important examples include:
1. The algebra C(X) of continuous complex-valued functions on a compact space
X, where involution is given by pointwise complex conjugation.
2. Any norm-closed, self-adjoint subalgebra of B(H) (the bounded operators on a
Hilbert space H) is a C ∗ -algebra with the induced norm and involution. Examples include the algebra K(H) of compact operators when H is infinitedimensional, or the algebra Mn of n × n complex matrices when H = Cn .
A particularly interesting class of C ∗ -algebras are crossed products, which are C ∗ algebras built from topological dynamical systems (see Appendix C). Given a system
(X, α), the associated crossed product C(X) ⋊α Z is a C ∗ -algebra that encodes the
space X as well as the action of α; C(X) ⋊α Z is a universal C ∗ -algebra containing
both a copy of C(X) and a unitary u such that uf u∗ = f ◦ α−1 for all f ∈ C(X).
This unitary renders the formerly outer automorphism f 7→ f ◦ α−1 of C(X) inner.
The radius of comparison of a C ∗ -algebra A, denoted rc(A), is a real-valued dimension theory which was introduced in [51, Definition 6.1] as a noncommutative
analogue of the covering dimension of a space. For example:
1. If X is compact, then rc(C(X)) ≈ 12 dim(X) [15, Theorem 1.1].
2. If α is an n-periodic homeomorphism of S 1 , then rc(C(S 1 ) ⋊α Z) ≈
dim(S 1 )
.
2n
In
particular, the radius of comparison sees both the dimension of the underlying
space and the periodicity of the homeomorphism.
The motivation for the radius of comparison comes from Atiyah-Hirzebruch Ktheory for smooth manifolds, which is built on the study of vector bundles. By
the Serre-Swam theorem, vector bundles over a compact manifold X are equivalent
5
to finitely-generated projective C ∞ (X)-modules. This can be generalized to projective A-modules (where A is a C ∗ -algebra) to gather cohomological invariants of the
C ∗ -algebra even when there are no vector bundles in the picture. The radius of
comparison measures the degree to which equivalence of A-modules is determined by
rank, detecting unstable homotopy phenomena in the noncommutative setting (as
discussed in Section 3.2).
1.3
Connections Between the Two and Summary of Results
The underlying goal of this dissertation is to verify Conjecture 1: that the mean
dimension of a minimal dynamical system is equal to twice the radius of comparison
of the corresponding crossed product. If true, this would allow for a new dimension
theory for dynamical systems, the dynamic dimension, defined by ddim(X, α) =
rc(C(X) ⋊α Z). This new invariant would extend the mean dimension in the case of
minimal systems, and provide an alternative invariant for systems in which the mean
dimension gives little or no information (such as finite-dimensional, finite-entropy,
and periodic systems).
The connection between the radius of comparison and the mean dimension was
first observed by Giol and Kerr in [18], who specifically constructed a dynamical
system so that the corresponding crossed product C ∗ -algebra would have nonzero
radius of comparison. In so doing, they observed that the system also had nonzero
mean dimension. This connection was confirmed by Lin and Phillips, who verified
Conjecture 1 in the case of (finite-dimensional) smooth manifolds.
Theorem. [32] If X is a smooth manifold and α is a minimal diffeomorphism, then
mdim(X, α) = 2 rc(C(X) ⋊α Z) = 0.
This result was later generalized by Toms and Winter to all finite-dimensional
topological spaces.
6
Theorem. [55, Theorems A & B] If (X, α) is a finite-dimensional minimal system,
then
mdim(X, α) = 2 rc(C(X) ⋊α Z) = 0.
In Chapters 5-7, we extend these results to infinite-dimensional systems by investigating several important cases. In the case of mean dimension zero systems, we
have verified Conjecture 1 under a regularity assumption known as Z-stability:
Theorem 1.1. If (X, α) is a minimal system with mdim(X, α) = 0, then
rc(C(X) ⋊α Z) = 0 if and only if C(X) ⋊α Z is Z-stable.
This result is particularly important in the context of the classification program
for C ∗ -algebras, giving evidence that crossed products of mean dimension zero systems are amenable to classification. Virtually no information is known about the
classification of crossed products coming from infinite-dimensional systems, and this
result is encouraging evidence.
It is important to note that the mean dimension zero case is not restricted to finitedimensional systems, and so 1.1 does provide information on some infinite-dimensional
systems and is indeed a generalization of [55, Theorems A & B]. In Chapter 4, we
show that a generic family of minimal, infinite-dimensional, mean dimension zero
systems exist. What is more, such systems are not constructed as inverse limits of
finite-dimensional systems, so the techniques used in the finite-dimensional case do
not work for general mean dimension zero systems.
In the more general case (for systems with positive mean dimension) we partially
verify Conjecture 1 by an inequality using a Cantor minimal system as a spacing
device:
Theorem 1.2. Suppose (X, α) is any minimal system and (Y, β) is a Cantor minimal
system such that (X × Y, α × β) is minimal. Then
rc(C(X × Y ) ⋊α×β Z) ≤
1
1
mdim(X × Y, α × β) = mdim(X, α).
2
2
7
This result, proved in Chapter 6, gives a large new class of infinite-dimensional
systems that provide evidence for Conjecture 1. What is more, Theorem 1.2 is the
first result verifying Conjecture 1 in the case of positive mean dimension, which used
to be completely unknown. In Chapter 4, we give examples showing that the class of
minimal systems which satisfy the conditions of Theorem 1.2 (that is, their product
with a Cantor minimal system is minimal) is large.
In Chapter 7, we verify the reverse inequality to Theorem 1.2 for Giol-Kerr systems:
Theorem 1.3. Suppose (X, α) is a Giol-Kerr system. Then
rc(C(X) ⋊α Z) ≥
1
mdim(X, α).
2
Since these systems are the only minimal systems known to give crossed products
with positive radius of comparison, this represents a fundamental test case of Conjecture 1. Additionally, we show that rc(C(X ×Y )⋊α×β Z) = 21 mdim(X ×Y, α×β) when
(X, α) is a Giol-Kerr system and (Y, β) is a Cantor odometer, verifying Conjecture 1
for this family of systems.
8
CHAPTER 2. Topological Dynamical Systems and the Mean
Dimension
In this chapter, we give an overview of the mean dimension for topological dynamical
systems and a closely related concept called the small boundary property. Recall
that a topological dynamical system is simply a pair (X, α) where X is a compact
metrizable space and α is a homeomorphism of X (also called the action on the space).
See Appendix B for a more complete introduction. Many dimensional invariants for
dynamical systems exist, the simplest being the covering dimension of X.
Definition 2.1. Let X be a topological space. The covering dimension of X (also
called the topological or Hausdorff dimension) is the smallest integer d such that every
finite open cover of X can be refined so that each point in X is contained in at most
d + 1 elements of the cover.
This definition agrees with the intuition of dimension in the sense that the covering
dimension agrees with the affine dimension of a finite simplicial complex. However,
this dimension theory is unsatisfying for dynamical systems, since it ignores the action. For this reason, the mean dimension was developed as an ‘equivariant’ covering
dimension.
2.1
Mean Dimension
Definition 2.2. Let X be a compact metrizable space, and let U be a finite open
cover of X. The order of the cover U is defined by
ord(U) = sup
x∈X
X
U ∈U
χU (x) − 1.
(2.1)
9
Colloquially, ord(U) counts the maximum number of overlaps among the elements of
U. Recall that a cover V refines U, written V ≺ U, if every V ∈ V is a subset of some
U ∈ U. The dimension of the cover U is defined by
D(U) = inf ord(V).
V≺U
(2.2)
Using this notation, we can re-define dim X = supU D(U).
Remark 2.1. Recall that every finite open cover U of a space X defines a finite
simplicial complex K(U) called the nerve of U. The vertices of K(U) correspond to
the cover elements {[U ]}U ∈U , and a face exists between the vertices [U0 ], [U1 ], . . . , [Un ]
if ∩i Ui is nonempty. Intuitively, K(U) forms a finite simplicial model of X. With this
in mind, observe that ord(U) = dim(K(U)).
The idea of mean topological dimension, first proposed by Gromov, was used by
Lindenstrauss and Weiss in [34] to study the problem of when a minimal system
can be embedded into the full shift on the Hilbert cube. Roughly speaking, ‘small
and slowly-moving’ systems have small mean dimension while ‘large and turbulent’
systems have large mean dimension. In particular, the mean dimension of any dynamical system on a finite-dimensional space has mean dimension zero, as does any
topological dynamical system with finite entropy (see [34, p.14]). The definition of
mean dimension (alternatively, see [34, Definition 2.6]) follows:
Definition 2.3. The mean dimension of a dynamical system (X, α) is given by
1
D U ∨ α−1 (U) ∨ · · · α−n+1 (U)
n→∞ n
mdim(X, α) = sup lim
U
where U ∨ V = {U ∩ V }U ∈U ,
(2.3)
V ∈V .
Example 2.1.
1. If X has finite covering dimension, then mdim(X, α) = 0 for any map α. This
follows because, as noted after Equation 2.2, D(U) ≤ dim(X) for all U. However, the converse is not generally true. There do exist infinite-dimensional
systems with mean dimension zero, as shown in Chapter 4.
10
2. If α is the full shift on X = K Z , where K is a finite simplicial complex, then
mdim(X, α) = dim(K) by [34, Proposition 3.1].
3. For any n ≥ 1, mdim(X, αn ) = n mdim(X, α) by [34, Proposition 2.7].
4. The trivial (identity) action on any space X has mean dimension zero: for any
cover U, the joins U ∨ U ∨ · · · ∨ U will eventually stabilize (after no more than
2|U | iterations) so the limit n1 D(U ∨ · · · ∨ U) will be zero for any U.
5. If X = Y n for some space Y , cyclic shift σn : X → X defined by σn (y1 , . . . , yn ) =
(yn , y1 , . . . , yn−1 ) has mean dimension zero, since σnn is the identity (combining
3. and 4.).
6. If (X, α) and (Y, β) are any two systems, then
mdim(X × Y, α × β) ≤ mdim(X, α) + mdim(Y, β)
by [34, Proposition 2.8].
Remark 2.2. The original definition of mean dimension (see [21, Section 1.5.]) is
equivalent, but uses slightly different terminology. Since both definitions are useful,
we include it here.
Let X be a compact space, and fix a metric d on X. For any ǫ > 0, we define the
width dimension
Widimǫ (X, d) = inf
)
There exists an ǫ-embedding X → K
dim(K) .
where K is a finite simplicial complex.
(
(2.4)
By ǫ-embedding, we mean any continuous map f : X → K such that diamd (f −1 (k)) <
ǫ for all k ∈ K. The width dimension is a more malleable version of covering dimension, the idea being that Widimǫ (X, d) = n if X can be embedded into an
n-dimensional simplicial complex ‘up to ǫ.’ In particular, if X has finite covering
dimension, then limǫ→0 Widimǫ (X, d) = dim(X).
Now if α is a homeomorphism of X, we can define the sequence of metrics dn on
X by dn (x0 , x1 ) = sup{d(αi (x0 ), αi (x1 )) | 0 ≤ i < n}. Note that this sequence is ‘increasing’ in the sense that dn (x0 , x1 ) ≤ dn+1 (x0 , x1 ) for any x0 , x1 ∈ X. Furthermore,
11
dn (x0 , x1 ) is large if α moves the points x0 and x1 far apart, i.e., the metrics dn grow
more quickly the more ‘turbulent’ α is. The mean dimension can now be alternatively
defined by
1
Widimǫ (X, dn ).
ǫ→0 n→∞ n
mdim(X, α) = lim lim
(2.5)
One particular advantage of the width dimension definition is the ease with which
it can be generalized to actions of arbitrary amenable groups. In this case, the metrics
dn are replaced by dFn (x0 , x1 ) = supg∈Fn d(gx0 , gx1 ) where (Fn ) is a Følner sequence
for G. The mean dimension then becomes
mdim(X, G) = lim lim
ǫ→0 n→∞
2.2
1
Widimǫ (X, dFn ).
|Fn |
(2.6)
Orbit Capacity and the Small Boundary Property
Definition 2.4. [34, Definition 5.1-2] For any dynamical system (X, α), the orbit
capacity of a subset E ⊂ X is defined by
n−1
ocap(E) = lim sup
n→∞ x∈X
1X
χE (αk x).
n k=0
(2.7)
We call E small if ocap(E) = 0.
Example 2.2.
1. The orbit capacity has many measure-theoretic properties, for example, ocap(X) =
1, ocap(∅) = 0, and ocap(E ∪ F ) ≤ ocap(E) + ocap(F ).
2. If E is closed, then
ocap(E) = sup{µ(E) | µ is an α-invariant probability measure on X}
(see [33, p. 232]).
3. The orbit capacity can be used to introduce measure-theoretic techniques in the
continuous (topological) setting. For example, we have a continuous analogue
of the Rokhlin tower lemma (see Theorem B.1):
12
Lemma 2.1. [33, Corollary 3.4] Let (X, α) be a topological dynamical system.
Then for any ǫ > 0, there exists a continuous level function n : X → R such
that E = {x ∈ X | n(α(x)) 6= n(x) + 1 or n(x) 6∈ Z} has ocap(E) < ǫ.
Definition 2.5. We say that the system (X, α) has the small boundary property. If
for any x ∈ X and any neighborhood U of x, there exists a neighborhood x ∈ V ⊂ U
with small boundary (i.e. ocap(∂V ) = 0).
1
In [33, Theorem 6.2] it was shown that the small boundary property is in fact
equivalent to mean dimension zero for nontrivial dynamical systems (specifically, extensions of minimal systems). A strengthened version of the small boundary property,
known as the topological small boundary property (see [8, Definition III.5]) was used
in [8] to show that smooth finite-dimensional uniquely ergodic minimal systems have
the dynamic comparison property, a purely topological analogue of strict comparison
(see [8, Definition III.18]).
2.3
Flaws in the Mean Dimension
Although the mean dimension is a useful measurement, it has several inherent
weaknesses. First of all, the mean dimension cannot distinguish between two finitedimensional systems (no matter how different their dynamics) since every finitedimensional system has mean dimension zero. Secondly, as discussed in [33], the
mean dimension breaks down in the presence of periodic points. Although periodic
points represent a common dynamical obstruction (for example, if a system has too
many periodic points then it cannot be embedded into a shift system) the mean
dimension does not capture this phenomena.
Fortunately, the radius of comparison provides an alternative invariant which does
not share these problems. Defining the dynamic dimension of a dynamical system as
ddim(X, α) = rc(C(X) ⋊α Z), we can use the radius of comparison as an alternative
1
A zero-dimensional space is one where every point has a neighborhood with empty boundary. In
this way, the small boundary property is the dynamic analogue of ‘zero-dimensional.’
13
dimension theory. As stated in the Introduction, the goal of this dissertation is to show
that the radius of comparison agrees with the mean dimension for minimal systems
(where the mean dimension works well), and provides a more nuanced measurement
of finite-dimensional and periodic systems (where the mean dimension breaks down).
To illustrate this point, we give the following two examples.
Example 2.3. Let X be a finite-dimensional space and let (X, id) denote the identity
system on X. Of course, mdim(X, id) = 0 since dim X is finite, but in general any
space under the identity action has zero mean dimension. On the other hand, we know
by Example C.2.1 that C(X) ⋊id Z ∼
= C(X × S 1 ) and hence ddim(X, id) ≈
dim X+1
2
by Theorem 3.2. That is, the radius of comparison picks up the dimension of the
underlying space even though the mean dimension is zero.
Example 2.4. Let X be a finite-dimensional space and let (X, α) be an n-periodic
n−1
system, in the sense that αn = id and X has a decomposition X = ⊔i=0
Xi where
α(Xi ) = Xi+1
(mod n)
(and hence each Xi ∼
= X0 ). In this case, we still have mdim(X, α) =
0 by finite-dimensionality. However, the crossed product C(X) ⋊α Z is isomorphic to
Mn C(X0 × S 1 ) as shown in Corollary 3.13. Hence ddim(X, α) ≈
dim X+1
2n
by Theorem
3.2. In this case, the radius of comparison sees both the dimension of the space and
the periodicity of the action even though the mean dimension does neither.
14
CHAPTER 3. Crossed Products and the Radius of
Comparison
In this chapter we define the radius of comparison of a C ∗ -algebra, with special focus
on the radius of comparison of crossed products. Additionally, we collect and review
the structure theorems about crossed products we will need in order to prove our
main results in Chapters 5, 6, and 7.
This chapter is organized as follows: in Section 3.1, we define the radius of comparison and give several important examples. In Section 3.2, we give a geometric
interpretation of the radius of comparison and exhibit geometric tools which can be
used to compute it. In Section 3.3, we focus on the radius of comparison for crossed
products and compute several important example cases. Finally, in Section 3.4, we
review the classification of crossed products and the structure theorems we will need
to prove our main results.
3.1
The Radius of Comparison
Definition 3.1. Let A be a C ∗ -algebra and Cu(A) its Cuntz semigroup. As shown
in Appendix D, any 2-quasitrace τ ∈ QT2 (A) induces a state dτ : Cu(A) → [0, ∞]
defined by dτ ([a]) = limn τ (a1/n ). For any a, b ∈ M∞ (A)+ , a . b implies dτ ([a]) ≤
dτ ([b]). If conversely dτ ([a]) < dτ ([b]) for all τ ∈ QT2 (A) implies that a . b, we say
that A has strict comparison of positive elements.
More generally, we say A has r-comparison (for r ≥ 0) if dτ ([a]) + r < dτ ([b]) for
all τ ∈ QT2 (A) implies that a . b. We define the radius of comparison rc(A) as the
infimum over all such r, with the convention that rc(A) = ∞ if no such r exists.
15
Example 3.1.
1. Suppose A = C. The usual trace τ (a) =
1
n
P
i
ai,i is the only quasitrace on
Mn (C). Consider the induced state on Cu(C): for any a ∈ Mn (C)+ ,
dτ (a) = lim τ (a1/k ) = lim
k
k
1 X 1/k
1 X 1/k
(a )i,i = lim
ai,i = rank(a).
k n
n i
i
Since a . b if and only if rank(a) ≤ rank(b) (see Example D.1), it follows that
a . b if and only if dτ (a) ≤ dτ (b). Thus A = C has strict comparison.
2. For more general C ∗ -algebras, the rank of a positive element a ∈ M∞ (A) is the
function rank(a) : QT2 (A) → R defined by rank(a)(τ ) = dτ (a) (notice that this
extends the usual definition of rank for positive matrices). With this notation,
we can equivalently define strict comparison as the property that a . b if and
only if rank(a) ≤ rank(b). In this sense, the radius of comparison measures the
degree to which (sub)equivalence of positive elements is determined by rank, a
concept further explored in Section 3.2.
3.2
The Radius of Comparison from a Geometric Perspective
The definition of the radius of comparison given in Section 3.1 is completely de-
termined by the Cuntz semigroup of a C ∗ -algebra, which obscures the geometric
motivation for the invariant. An heuristic interpretation of Definition 3.1 is that
the radius of comparison of a C ∗ -algebra A measures the degree to which the Cuntz
equivalence class of a positive element is determined by its trace(s). What we show in
this section is that this definition has a more geometric interpretation: the radius of
comparison measures the degree to which the equivalence of A-modules is determined
by rank, detecting unstable homotopy phenomena in the noncommutative setting.
Consider the case of a commutative C ∗ -algebra A = C(X), where X is a smooth
manifold. As shown in Appendix D the Murray-von Neumann semigroup V (A), from
which K0 (A) is constructed, sits naturally inside the Cuntz semigroup Cu(A). Rather
than consider V (A), we can use the topological perspective explained in Appendix
16
A to investigate the (isomorphic) semigroup Vect(X), and the topological K-theory
group K 0 (X). Just like K0 (A), the group K 0 (X) has a partial order defined by
[p]0 ≤ [q]0 if p is (stably isomorphic to) a sub-bundle of q. As shown in the following
Theorem, this partial order is intimately related to the topological properties of X:
Theorem 3.1. [24, Chapter 9, Theorems 1.2 & 1.5] If rank(p) + ⌊ 12 dim(X)⌋ ≤
rank(q), then [p]0 ≤ [q]0 .
This geometric condition motivates the definition of the radius of comparison as
a noncommutative covering dimension. Note that any trace τ : C(X) → C induces
a map K0 (C(X)) → K0 (C) = Z. In the commutative case, all traces are (densely)
spanned by the point evaluations, which simply return a projection’s rank. Hence
Theorem 3.1 can be rephrased for elements [p]0 , [q]0 ∈ K0 (A) as follows: if τ (p) +
1
2
dim(X) ≤ τ (q) for all traces τ on C(X) then [p]0 ≤ [q]0 . Compare this to the
definition of the radius of comparison, which states that rc(A) = r if for all [a], [b] ∈
Cu(A), dτ ([a]) + r ≤ dτ ([b]) implies [a] . [b]. With this perspective in mind, Theorem
3.1 can be extended to the Cuntz semigroup:
Theorem 3.2. [15, Theorem 1.1], [51, Theorem 6.6] Let X be a compact CWcomplex. Then rc(C(X)) ≤ 12 dim X. Furthermore:
1. If dim X is odd, then rc(C(X)) = max{0, dim 2X−1 − 1}.
2. If dim X is even, then
dim X
2
− 2 ≤ rc(C(X)) ≤
dim X
2
− 1.
Throughout this dissertation, we summarize these facts using the notation
rc(C(X)) ≈ 21 dim(X).
To extend the geometric perspective from smooth manifolds to the noncommutative case, first recall the Serre-Swan theorem, which states that vector bundles over X
are equivalent to finitely-generated projective C ∞ (X)-modules. In the definition of
the Cuntz semigroup (particularly, Definition D.9) this is generalized to projective Amodules, with the aim of gathering cohomological invariants of a (noncommutative)
C ∗ -algebra A, even when the notion of vector bundle does not apply.
17
We conclude this section with an important example which uses topological KTheory and characteristic classes (described in Appendix A) to give an explicit example of a commutative C ∗ -algebra that does not have strict comparison.
Example 3.2. Let A = C(X), where X is the cube [−1, 1]6 . To show that rc(A) 6= 0,
we construct two positive elements a and b in M∞ (C(X)) such that rank(a) < rank(b)
but [a]Cu 6≤ [b]Cu . We do this using cohomological properties of X.
Although X is contractible, it contains an embedded copy of the spheres Y =
S 2 × S 2 . The 2-sphere has non-trivial cohomology (specifically, H even (S 2 ; Z) = Z ⊕ Z)
and the cohomological obstructions from S 2 can be lifted to obstructions inside Cu(A).
Identifying S 2 with CP1 , define a rank-one vector bundle b over S 2 by
E = {(v, y) ∈ CP1 × S 2 | v ∈ y} and b : E → S 2 the projection onto the second coordinate (the Bott bundle). Then [b]0 is a nontrivial element of K 0 (S 2 ), which can be
seen by computing ch1 ([b]0 ) 6= 0 ∈ H even (S 2 ; Z) (see Appendix A for a brief review of
characteristic classes). Using Corollary 7.1, this implies that [t1 ]0 6≤ [b × b]0 in K 0 (Y ),
where t1 : C × Y → Y is the rank-one trivial bundle over Y . Set p = t1 and q = b × b.
We now extend p and q to the desired elements in M∞ (C(X))+ . In general,
extending vector bundles over a subspace to positive elements in the whole algebra is
not a straightforward matter (see e.g. [53, Lemma 3.10]) but the general argument is
not difficult to understand. Let U be an open subset of X of which Y is a deformation
retract. Since deformation retracts induce isomorphisms on K-theory, p and q can
be extended to vector bundles p̃ and q̃ over U . By the Tietze extension theorem,
there exists a function f : X → [0, 1] such that f |Y = 1 and f |X\U = 0. Now simply
define a = f p̃ and b = f q̃ + (1 − f )t̃2 , where t̃2 denotes the trivial rank-two bundle
over X. Since a and b were constructed so that rank(a) ≤ 1, rank(b) ≥ 2, and yet
[a |Y ] = [p] 6≤ [q] = [b |Y ], we have our desired elements.
Remark 3.1. Example 3.2 contains an important insight into the difference between
the K0 -group and the Cuntz semigroup. While the K0 -group contains cohomological
information about X, the Cuntz semigroup carries cohomological information not
18
just about X but also every closed subspace of X. This gives a glimpse into how
complicated the Cuntz semigroup can be even in the commutative case.
3.3
The Radius of Comparison of a Crossed Product
Although the definition of the radius of comparison given in Definition 3.1 applies
to all C ∗ -algebras, the algebras we investigate in this dissertation are crossed products
coming from topological dynamical systems. In this section, we focus on the Cuntz
semigroup and the radius of comparison in this case. Throughout, let (X, α) be a
topological dynamical system.
The algebra C(X) ⋊α Z contains a canonical copy of C(X) as a commutative subalgebra, so by Example D.1.4 we have a natural morphism Cu(C(X)) → Cu(C(X) ⋊ Z),
although this map is not injective in general. A natural place to begin our investigation of Cu(C(X) ⋊ Z) is with this commutative subalgebra. The order structure
on Cu(C(X)) is illuminated by the following result, and we include a short proof to
exhibit the techniques we will use later in this section.
Lemma 3.3. [4, Proposition 2.5] Suppose f, g ∈ C(X) are positive functions. Then
f .C(X) g if and only if supp(f ) ⊂ supp(g) (throughout this dissertation, supp(f )
denotes the open set {f > 0}).
Proof. If f . g then for some sequence hn ∈ C(X) we have hn g h̄n → f , so clearly
supp(f ) ⊂ supp(g). Thinking of this another way, we know by Lemma D.2 that
for any positive elements a, b ∈ M∞ (C(X)), a . b implies dτ ([a]) ≤ dτ ([b]) for all
traces τ on C(X). In particular, if τx is the evaluation trace at x ∈ X, dτx ([a]) =
rank(a)(x). For f ∈ C(X), we have rank(f )(x) = χsupp(f ) (x) and so clearly f . g
implies supp(f ) ⊂ supp(g) by rank considerations.
The converse argument we copy from [4, Proposition 2.5]. Assume f, g ∈ C(X)
with supp(f ) ⊂ supp(g). Since f ∼ f 2 and g ∼ g 2 by [4, Corollary 2.6], it suffices to
show that f 2 . g 2 . For ǫ > 0, define the compact subset K ⊂ X by K = {f ≥ ǫ}.
Clearly each K ⊂ supp(g), and since K is compact there exists δ > 0 such that g > δ
19
on K. Define U = {g > δ}, which is an open neighborhood of K. Now use Urysohn’s
Lemma to construct a function e : X → [0, 1] such that e |K = 1 and e |X\U = 0. Then
kf 2 −
ef 2 ef
g gk
g
< ǫ and so f 2 . g 2 .
Viewing C(X) inside C(X) ⋊α Z, Lemma 3.3 can be extended to crossed products
as follows.
Lemma 3.4. Let f, g ∈ C(X) ⊂ C(X) ⋊α Z. Suppose that there exists a finite family
{Ui } of open subsets of X such that supp(f ) ⊂ ∪i Ui and integers {ni } such that all
αni (Ui ) are pairwise disjoint subsets of supp(g). Then f .C(X)⋊Z g.
Proof. Let {Ui } and {ni } be given, and define functions {fi } with supp(fi ) = Ui . By
P
fi . Therefore, f .C(X) ⊕fi by [4, Lemma 2.10] so clearly
Lemma 3.3, f .C(X)
f .C(X)⋊Z ⊕fi . Next, defining the unitary u = ⊕uni , we have ⊕fi ∼ u(⊕fi )u∗ =
⊕fi ◦ α−ni . But the fi ◦ α−ni are orthogonal, since supp(fi ◦ α−ni ) = αni (Ui ). So
P
fi ◦ α−ni .C(X)⋊Z g.
⊕fi ◦ αni ∼C(X)⋊Z
X
αn1 (U1)
αn2 (U2)
U1
U2
supp(f )
supp(g)
Figure 3.1.: Dynamic comparison in Lemma 3.4.
20
A very similar idea was studied by J. Buck in [8], who defined an analogue of
strict comparison in purely dynamical terms.
Definition 3.2. [8, Definition III.18] Suppose (X, α) has a unique ergodic measure
µ. We say that (X, α, µ) has the dynamic comparison property if whenever U ⊂ X
is open and C ⊂ X is closed with µ(C) < µ(U ), then there exists a finite partition
of unity {fi } for C and integers {ni } such that {supp(fi ◦ αni )} are pairwise disjoint
subsets of U .
Observation 3.1. An immediate corollary of Lemma 3.4 is that if (X, α, µ) has the
dynamic comparison property, µ(supp(f )) < µ(supp(g)) implies f .C(X)⋊Z g.
The conditions of Lemma 3.4 can be difficult to arrange, but in essence we require
that the support of f is ‘equivariantly smaller’ than the support of g (allowing us to
malleate supp(f ) with the dynamics of the system). If we asymptotically extend this
condition, it bears a striking resemblance to the orbit capacity in Definition 2.4. We
summarize this idea in the following conjecture:
Conjecture 2. Let f, g ∈ C(X) ⊂ C(X) ⋊α Z. If ocap(supp(f )) ≤ ocap(supp(g))
then f .C(X)⋊Z g.
Determining the Cuntz equivalence classes of arbitrary positive elements over
C(X) (not to mention C(X) ⋊α Z) is not so simple, but the following fundamental
result proves that the ordering is determined largely by rank.
Theorem 3.5. [53, Theorem 3.15] Let X be a compact, finite-dimensional, metrizable
space. If a, b ∈ Mn (C(X)) and
rank(a)(x) + 9 dim X ≤ rank(b)(x) for all x ∈ X
then a . b.
Theorem 3.5 motivates the definition of the radius of comparison, and was the first
result bounding the radius of comparison of a commutative C ∗ -algebra, answering
21
Question 6.9 from [51]. As shown in Section 3.2, the inequality in Theorem 3.5 can
be tightened so that in general, rc(C(X)) ≈ 12 dim X. As an easy first example, this
lets us compute the radius of comparison for trivial crossed products.
Example 3.3. As shown in Appendix C, if id denotes the identity action
on a space X, then C(X) ⋊id Z ∼
= C(X × S 1 ).
Therefore, by Theorem 3.2,
rc(C(X) ⋊id Z) ≈ 21 (dim X + 1).
To extending results about Cu(C(X)) to Cu(C(X)⋊Z), we include several results
from [7] that describe what positive elements in C(X) ⋊α Z look like and define useful
maps for approximating C(X) ⋊α Z using C(X).
Lemma 3.6. [7, Corollary 4.1.6] The element
P
i
fi ui ∈ Cc (Z, C(X)) is positive
in C(X) ⋊α Z if and only if for any finite sequence n1 , . . . , nk ∈ Z the matrix
(fni −nj ◦ αni )i,j is positive in Mk C(X).
Lemma 3.7. [7, Lemma 4.2.2] For every finite set F ⊂ Z and subset {fi }i∈F ⊂ C(X),
!∗
!
X
X
X
fi ui
fj uj =
(f¯i fj ) ◦ αi uj−i .
(3.1)
i∈F
In particular,
j∈F
P
i,j∈F (f̄i fj )
i,j∈F
◦ α−i ui−j is positive in C(X) ⋊α Z.
Lemma 3.8. [7, Lemma 4.2.3] For every finite set F ⊂ Z, there exist c.p.c. maps
ψ
ϕ
F
F
C(X) ⋊α Z −→
MF (C(X)) −→
C(X) ⋊α Z,
defined by ψF (f un ) =
P
i∈F ∩F +n (f
such that ϕF ◦ ψF (f un ) =
◦ αi ) ⊗ ei,i−n and ϕF (f ⊗ ei,j ) =
1
(f
|F |
◦ α−i )ui−j ,
|F ∩F +n|
f un .
|F |
We finish this section by computing the structure and radius of comparison in
several important example cases: the crossed product corresponding to a full shift,
and the crossed product of a periodic dynamical system. The trivial shift is an easy
first case.
22
Example 3.4. Consider the full shift on the one-point space, X = {∗}Z = {∗}. The
action of Z on X is trivial, hence C(X) ⋊σ Z ∼
=C⋊Z∼
= C ∗ (Z) ∼
= C(S 1 ). It is well
known (see, e.g. [15]) that rc(C(S 1 )) = 0 = 12 dim{∗}.
To compute the radius of comparison of a general shift system we need the following lemma.
Lemma 3.9. Let A and B be C ∗ -algebras, and suppose we have c.p.c. order zero
ψ
ϕ
maps A → B → A such that Cu(ϕ ◦ ψ) : Cu(A) → Cu(A) is the identity map. Then
rc(A) ≤ rc(B).
Proof. First,
note
that
the
maps
Cu(ψ) : Cu(A) → Cu(B)
and
Cu(ϕ) : Cu(B) → Cu(A) are well-defined, and Cu(ϕ) ◦ Cu(ψ) = Cu(ϕ ◦ ψ)
by [60, Corollary 3.5]. For constant 0 ≤ r < rc(A), find elements a, a′ ∈ (A ⊗ K)+
such that dτ ([a]) + r < dτ ([b]) for all quasitraces τ ∈ QT2 (A) but a 6. a′ . Then it
also follows that ψ(a) 6. ψ(a′ ), since ψ preserves Cuntz equivalence by [60, Corollary
3.5]. Furthermore, by [60, Corollary 4.4], the map ψ ∗ : QT2 (B) → QT2 (A) given by.
ψ ∗ (τ ) = τ ◦ ψ is well-defined. Hence for all τ ∈ QT2 (B),
dτ (ψ(a)) + r = dψ∗ τ (a) + r
< dψ∗ τ (a′ )
= dτ (ψ(a′ )).
Thus rc(B) ≥ r, and hence rc(B) ≥ rc(A).
Corollary 3.10. Let K be a finite simplicial complex, and denote by σ the full shift
on K Z . Then
rc(C(K Z ) ⋊σ Z) ≥ rc(C(K)) ≈
1
dim K.
2
Proof. Set A = C(K Z ) ⋊σ Z. Define the map π : K Z → K by projection onto the
zeroth coordinate. Composing the homomorphism π ∗ : C(K) → C(K Z ) with the inclusion C(K Z ) → A gives us the c.p.c. order zero map (in fact, a homomorphism)
ψ : C(K) → A.
23
Now define the map d : K → K Z as the diagonal embedding k 7→ (. . . , k, k, . . . ).
This induces a homomorphism d∗ : C(K Z ) → C(K). Furthermore, the map d is an
equivariant map from the identity system (K, idK ) (as an action of the trivial group)
to (K Z , σ) since d ◦ idK = σ ◦ d. Hence d∗ extends to a homomorphism ϕ : A → C(K)
by the universal property of crossed products.
It only remains to check that ϕ◦ψ is the identity at the level of Cuntz semigroups.
d
π
This is clear by construction, since the maps K → K Z → K satisfy π◦d = idK . Hence
for any function f ∈ C(K), ϕ(ψ(f )) = ϕ(f ◦ π) = f ◦ π ◦ d = f .
To compute the radius of comparison of a periodic crossed product, we first prove
the following decomposition results. These will also be useful in Chapters 5, 6, and 7
when we decompose arbitrary dynamical systems into cyclic (or almost-cyclic) subsystems.
Theorem 3.11. [20, Theorem 4.1] Let G be a locally compact group acting on X and
let H be a closed subgroup of G. Suppose there exists a continuous map π : X → G/H
that is G-equivariant (where G acts on G/H by translation) and set Y = π −1 (H).
Then any measurable section G/H → G induces an isomorphism
C(X) ⋊ G ∼
= K(L2 (G/H)) ⊗ (C(Y ) ⋊ H).
Corollary 3.12. Let (X, α) be a dynamical system and suppose X has a decomn−1
position into disjoint closed subsets X = ⊔k=0
Xk such that α(Xk ) = Xk+1
(mod n) .
Then
C(X) ⋊α Z ∼
= Mn (C(X0 ) ⋊αn Z).
Proof. Consider nZ ⊂ Z as a closed subgroup, and define the map π : X → Z/n by
π(Xi ) = i. This map is continuous by the Pasting Lemma (since each Xi is closed
and disjoint) and equivariant by assumption. Notice that π −1 (0) = X0 . There exist
many continuous sections Z/n → Z, so applying Theorem 3.11 gives the isomorphism
C(X) ⋊α Z ∼
= K(L2 (Z/n)) ⊗ (C(X0 ) ⋊α nZ) ∼
= Mn (C(X0 ) ⋊αn Z).
24
Corollary 3.13. Suppose (X, α) is a dynamical system such that αn = idX and X has
n−1
a cyclic decomposition into disjoint closed subspaces X = ⊔k=0
Xk with α(Xk ) = Xk+1
(mod n). Then
C(X) ⋊α Z ∼
= Mn (C(X0 × S 1 )).
Proof. By Corollary 3.12, we have C(X) ⋊α Z ∼
= Mn (C(X0 ) ⋊αn Z). With the added
assumption that αn = id, it follows that C(X0 ) ⋊αn Z ∼
= C(X0 × S 1 ) (see Example
C.2.1).
Example 3.5. Suppose (X, α) is an n-periodic dynamical system with decomposition
n−1
Xk as described in Lemma 3.13. Then C(X) ⋊α Z ∼
X = ⊔k=0
= Mn (C(X0 × S 1 )) and
hence rc(C(X) ⋊α Z) = rc(Mn (C(X0 × S 1 )) ≈
3.4
dim X+1
2n
by Theorem 3.2.
Classification of Crossed Products
One of the most significant results confirming Elliott’s conjecture (see Appendix
A.3) was given in [55], where it was shown that the crossed product C ∗ -algebra
associated to a minimal, finite-dimensional dynamical system is Z-stable (Definition
A.20). As a result, these algebras are classified by their graded ordered K-theory in
the uniquely ergodic case:
Theorem 3.14. [55, Theorem 0.1] Suppose A and B are crossed product C ∗ -algebras
coming from minimal, finite-dimensional, uniquely ergodic dynamical systems. If
there exists an isomorphism K∗ (A) → K∗ (B), then it lifts to a C ∗ -isomorphism A → B.
A major leap forward in the classification problem for crossed products was made
by Lin and Phillips in [31, 32], who use techniques developed by Putnam (see [44]) to
show that crossed products coming from minimal dynamical systems possess many
important large subalgebras which capture much of the structure of the full crossed
product. Furthermore, Lin and Phillips show that these large subalgebras can be
decomposed as the direct limit of recursive subhomogeneous algebras, providing an
essential tool for investigating their structure.
25
In Section 3.4.1, we give the definition of a recursive subhomogeneous algebra and
provide several important examples relating to crossed products. In Section 3.4.2, we
review large subalgebras and show how why they are so important to the classification
and study of crossed products.
3.4.1
Recursive subhomogeneous algebras
Definition 3.3. Given unital C ∗ -algebras A, B, and C, a surjective homomorphism
q : B → C, and a unital homomorphism ϕ : A → C, the pullback algebra (or restricted
sum) is defined by A ⊕C B = {(a, b) ∈ A ⊕ B | ϕ(a) = q(b)}.
A recursive subhomogeneous (RSH) algebra is a C ∗ -algebra that can be written as
an iterated pullback of subhomogeneous algebras of the form Mn C(X):
···
Mn0 C(X0 ) ⊕Mn1 C(Z1 ) Mn1 C(X1 ) ⊕Mn2 C(Z2 ) Mn2 C(X2 ) · · · ⊕Mnℓ C(Zℓ ) Mnℓ C(Xℓ ).
Example 3.6.
1. Pullbacks generalize ordinary direct sums, since if C = 0, A ⊕C B = A ⊕ B.
Hence subhomogeneous algebras of the form Mn1 C(X1 ) ⊕ · · · ⊕ Mnℓ C(Xℓ ) are
basic examples of RSH algebras.
2. Let X be a compact metrizable space, and suppose we have a closed cover X =
X0 ∪ X1 . By the Pasting Lemma, a pair of functions (f0 , f1 ) ∈ C(X0 ) ⊕ C(X1 )
define a function f ∈ C(X) if and only if f0 |X0 ∩X1 = f1 |X0 ∩X1 . That is to say,
we can write C(X) = C(X0 ) ⊕C(X0 ∩X1 ) C(X1 ) where C(X0 ) → C(X0 ∩ X1 ) and
C(X1 ) → C(X0 ∩ X1 ) are the restriction maps.
3. As shown in [31, Example 2.6], the prime dimension drop intervals (the building
blocks of the Jiang-Su algebra Z) are RSH algebras. Recall that
Ip,q = {f : [0, 1] → Mp ⊗ Mq | f (0) ∈ Mp ⊗ C1 and f (1) ∈ C1 ⊗ Mq }.
An alternate (RSH) definition of Ip,q is given by
Ip,q = (Mp C({0}) ⊕ Mq C({1})) ⊕(Mp ⊗Mq )⊕(Mp ⊗Mq ) Mpq C([0, 1]),
26
where the relevant maps are defined by:
Mp C({0}) ⊕ Mq C({1}) → (Mp ⊗ Mq ) ⊕ (Mp ⊗ Mq )
(a, b) 7→ (a ⊗ 1, 1 ⊗ b)
and
Mpq C([0, 1]) → (Mp ⊗ Mq ) ⊕ (Mp ⊗ Mq )
a 7→ (a(0), a(1)).
4. Locally trivial continuous fields over X with fiber Mn and NCCW-complexes
are other important examples of RSH algebras
Perhaps the most striking examples of RSH algebras appear as subalgebras of
crossed products. First studied by I. Putnam in [44], the orbit-breaking subalgebras
were later shown by N.C. Phillips and Q. Lin in [31] to be RSH.
Definition 3.4. [44, Section 3] Let (X, α) be a dynamical system and A = C(X) ⋊α Z
the corresponding crossed product. Denote by u ∈ A the canonical unitary implementing the action Z yα X. For any closed subset K ⊂ X, define the Putnam
algebra AK as the subalgebra of A generated by C(X) and uC0 (X \ K), that is,
AK = C ∗ (C(X), uC0 (X \ K)).
Example 3.7. Also called orbit-breaking subalgebras, Putnam algebras are extremely
useful for introducing cyclic structure to dynamical systems. As shown in Corollary 3.13, if a dynamical system (X, α) is n-periodic and we can decompose X
into a cycle of n disjoint closed subspaces X = K ⊔ α(K) ⊔ · · · ⊔ αn−1 (K), then
C(X) ⋊α Z ∼
= Mn (C(K × S 1 )).
Of course, this may not be the case. If αn is not the identity map then such a nice
decomposition does not exist in general. However, ‘breaking’ the action at αn−1 (K)
repairs this problem by ‘ignoring’ the image of αn−1 (K) under α. If we consider
AK = C ∗ (C(X), u(C(X \ K))) ∼
= C ∗ (C(X), uC(X \ αn−1 (K))) instead of A itself,
then we do get an isomorphism AK ∼
= Mn (C(K)).
27
Lemma 3.15. Let (X, α) be a dynamical system and set A = C(X) ⋊α Z. Suppose
n−1
X can be decomposed into a cycle of n disjoint closed subspaces X = ⊔i=0
Xi such
that α(Xi ) = Xi+1
(mod n) .
Then AXn−1 ∼
= Mn (C(X0 )).
Proof. Denote by un the element uχX\Xn−1 ∈ AXn−1 . Note that AXn−1 = C ∗ (C(X), un ).
For 0 ≤ i, j ≤ n − 1 define the elements Ei,j ∈ AXn−1 by
Ei,j = χXi ui−j
n .
(3.2)
Notice that
Ei,j Ek,ℓ = χXi uni−j χXk unk−ℓ
= χXi χXk+i−j
(mod n)
unk−ℓ+i−j
= δj,k χXk−j+i (mod n) uni−ℓ+k−j

 0 if j 6= k,
=
 E if j = k.
i,ℓ
By the universal property of matrix units, this induces a homomorphism Mn → AXn−1
that maps the standard matrix units ei,j 7→ Ei,j (here we index Mn from 0 to n − 1
for consistency of notation).
Define the map π : X → X0 by π |Xi = α−i , which induces a homomorphism
π ∗ : C(X0 ) → C(X).
Since C(X) is central in AXn−1 , we get a homomorphism
ϕ : Mn (C(X0 )) → AXn−1 such that ϕ(ei,j ⊗ f ) = Ei,j π ∗ (f ).
To see that ϕ is surjective, it suffices to show that its image contains un and
Pn−2
any f ∈ C(X). But this is clear, since un = ϕ( i=0
ei,i−1 ⊗ 1) and f =
P
n−1
ei,i ⊗ (f ◦ α−i ) |X0 ).
ϕ( n1 i=0
To show that ϕ is injective, we show that any homomorphism ψ : Mn (C(X0 )) → B
to a unital C ∗ -algebra B descends to a homomorphism γ : AXn−1 → B such that
γ ◦ ϕ = ψ. It suffices to define γ on generators. For f ∈ C(X), define γ(f ) =
Pn−1
Pn−2
ψ( n1 i=0
ei,i ⊗ (f ◦ α−i ) |X0 ), and define γ(un ) = ψ( i=0
ei,i−1 ⊗ 1). It is clear that
28
ψ = γ ◦ ϕ by definition. To see that γ is a homomorphism, it suffices to check that
γ(un )γ(f )γ(un )∗ = γ(un f u∗n ):
n−2
n−1
n−2
X
X
X
∗
−i
γ(un )γ(f )γ(un ) = ψ (
ei,i−1 ⊗ 1)(
ei,i ⊗ (f ◦ α ) |X0 )(
ei,i−1 ⊗ 1)∗
i=0
n−2
X
i=0
i=0
!
ei+1,i+1 ⊗ (f ◦ α−i ) |X0 )
= ψ(
i=0
n−1
X
= ψ(
ei,i ⊗ (f ◦ α−i−1 ) |X0 )
i=1
= γ(χX\Xn−1 (f ◦ α−1 ))
= γ(un f un ∗ ).
Example 3.8. Consider the special case where (X, α) is the 2-odometer (that is,
Q
X= ∞
i=1 Z/2 and α(x) = x+1 with ‘carryover’ addition, see Example B.4). Defining
K = {(xi ) ∈ X | x1 = · · · = xn = 0}, notice that X is the disjoint union of the 2n
n −1
subspaces K⊔α(K)⊔· · ·⊔α2
n
(K), with α2 (K) = K (although α is not 2n periodic).
Therefore, as explained in Lemma 3.15, AK ∼
= M2n (C(K)). As shown in [44, Theorem
3.3], since K is compact and totally disconnected, we can find a sequence of partitions
generating the topology of X, and define a corresponding chain of finite-dimensional
subalgebras whose union is dense in AK , showing that AK is AF. Of course, the general
case is more subtle. However, the ‘cyclic’ structure of Cantor minimal systems is key
to proof that the Putnam subalgebras have the structure of subhomogeneous algebras
of the form Mn (C(K)).
Theorem 3.16. [44, Theorem 3.3] Let (X, α) be a Cantor minimal system and set
A = C(X) ⋊α Z. Then for any compact subset K ⊂ X, AK is an AF-algebra.
Examples 3.7 and 3.8 show how orbit-breaking subalgebras are useful when a
crossed product can be decomposed into a disjoint cycle of subspaces. The problem
is that such cyclic decompositions are often not obvious in nature (and do not even
29
exist for connected minimal systems, for example). For this reason, we construct
almost-cyclic decompositions using Rokhlin towers.
Example 3.9. As shown in Lemma 3.15, if X can be written as a single Rokhlin
n−1 j
tower X = ⊔j=0
α (K), then AK = Mn (C(K)). More generally, suppose that X
i −1
breaks into ℓ Rokhlin towers, X = ⊔ℓi=1 ⊔nj=0
αj (Ki ), with K = ⊔i Ki . If we suppose
that the Ki are all closed, then X is the disjoint union of ℓ cycles. Each cycle
i −1 j
⊔nj=0
α (Ki ) corresponds to the subhomogeneous algebra Mni C(Ki ). Therefore, we
get an isomorphism AK ∼
= Mn1 C(K1 ) ⊕ Mn2 C(K2 ) ⊕ · · · ⊕ Mnℓ C(Kℓ ).
The problem, of course, is that the Rokhlin tower corresponding to K will not
consist of closed sets in general (see Definition B.4). Therefore the decomposition
of AK will not generally be a direct sum of algebras of the form Mni C(Ki ). It will,
however, be an RSH algebra consisting of these pieces, as shown in [31, Section 3]
(and also as a result of [30, Theorem 3]).
Theorem 3.17. [31, Section 3] Let (X, α) be a minimal system and set A = C(X) ⋊α Z.
If K ⊂ X is a closed set with nonempty interior, then AK is an RSH-algebra. Specifi −1
ically, if K induces the Rokhlin tower X = ⊔ℓi=1 ⊔nj=0
αj (Ki ) then
AK = · · · Mn1 C(K1 ) ⊕Mn2 C(Z2 ) Mn2 C(K2 ) ⊕Mn3 C(Z3 ) · · · ⊕Mnℓ C(Zℓ ) Mnℓ C(Kℓ )
where Zi = ∂Ki ∩ (K0 ∪ · · · ∪ Ki−1 ).
Since the Ki are not closed in general, it is necessary to consider their closures.
However, this means that the Rokhlin towers will no longer be disjoint. The most
complicated aspect of Theorem 3.17 is figuring out how the Ki ’s overlap. Any overlap
requires boundary conditions, which is the reason that the RSH structure is necessary
(see Example 3.6.2). Although the RSH-algebras AK may not be easy to describe, it
is nevertheless possible to prove useful results describing their structure:
Lemma 3.18. Let (X, α) be minimal and set A = C(X) ⋊α Z. If K ⊂ X is a
proper closed subset, then the inclusion map AK → A induces an isomorphism on
30
trace spaces. That is, T (AK ) ∼
= T (A) ∼
= Probα (X), the space of α-invariant probability measures on X.
Proof. Every trace on A corresponds to an α-invariant probability measure on X.
Since AK is by a subalgebra of A, every trace on A restricts to a trace on AK .
Conversely, a trace τ on AK = C ∗ (C(X), uC0 (X \ K)) defines a probability measure
µτ on X by restricting to C(X). Such a trace must satisfy τ (uf u∗ ) = τ (f ) for every
f ∈ C0 (X \K) and hence µτ (E) = µτ (α(E)) for any E ⊂ X \K (i.e. µτ is α-invariant
when restricted to X \ K). Since α is minimal by assumption, this implies that µτ is
α-invariant. Thus T (A) ∼
= T (AK ) ∼
= Probα (X).
Lemma 3.19. [40] Set A = C(X) ⋊α Z and let {fn } ⊂ C(X). The formal sum
P
n−1 j
n
n fn u is contained in AK if and only if fn = 0 on ∪j=0 α (K) for all n > 0 and
fn = 0 on ∪nj=1 α−j (K) for all n < 0.
Lemma 3.20. [40] If a =
PN
n=−N
fn un ∈ A = C(X) ⋊α Z is arbitrary and f ∈ C(X)
−1
n
vanishes on ∪N
n=−N α (K), then af, f a ∈ AK .
3.4.2
Large Subalgebras
Of particular interest among Putnam subalgebras of crossed products are the
maximal orbit-breaking subalgebras of the form A{x} , for a single point x ∈ X.
Discovered in [44, §3 and Theorem 4.1] and further developed in [30, 31, 39], these
large subalgebras are useful since many properties which hold for A{x} also hold for
A. For example: A{x} is simple (see [30, Proposition 12]), has the same K0 -group
as A ( [38, Theorem 4.1] and [43, Example 2.6]), and has the same trace simplex
( [30, Proposition 16]).
Furthermore, A{x} has a nice inverse limit decomposition. If K0 ⊃ K1 ⊃ · · · is a
descending chain of closed sets with nonempty interior such that ∩n Kn = {x}, then
A{x} = ∪n AKn = lim AKn . By Theorem 3.17, this gives A{x} as an inverse limit of
←−
RSH algebras.
31
Lemma 3.21. For any x ∈ X, the large subalgebra A{x} of A = C(X) ⋊α Z can be
written as an inverse limit of RSH algebras.
Because of these useful properties, the concept of a large subalgebra was made
precise in [40] for general simple C ∗ -algebras:
Definition 3.5. [40, Definition 2.2] A subalgebra B of a simple, separable, stably
finite, unital C ∗ -algebra A is large in A if:
1. B contains the unit of A,
2. B is simple,
3. the restriction map QT(A) → QT(B) is surjective, and
4. for all ǫ > 0, for any finite collection a1 , . . . , am ∈ A, for all v ∈ A+ with kvk = 1
and for all nonzero w ∈ B+ , there exists c1 , . . . , cm ∈ A and g ∈ B+ satisfying:
(a) 0 ≤ g ≤ 1,
(b) kcj − aj k < ǫ for all j = 1, . . . , m,
(c) (1 − g)cj , cj (1 − g) ∈ B for all j = 1, . . . , m, and
(d) g .Cu(B) w.
The definition of large subalgebras was designed to replicate the properties of the
orbit-breaking subalgebras A{x} ⊂ A = C(X) ⋊α Z. It is shown in [40, Theorem 3.8]
that these subalgebras are indeed large according to Definition 3.5:
Theorem 3.22. [40, Theorem 3.8] For any x ∈ X, A{x} is large in A = C(X) ⋊α Z.
Example 3.10. Suppose that (X, α) is the Cantor 2-odometer, as in Example 3.8,
and let x = (0, 0, 0, . . . ). As Example 3.8 shows, when Kn = {y ∈ X | y1 = y2 =
· · · = yn = 0}, AKn ∼
= M2n (C(Kn )). Clearly, we have ∩n Kn = {x}, and hence
A{x} = lim AKn = lim M2n (C(Kn )) = M2∞ (C).
−→
−→
32
Therefore, by Theorem 3.22, the 2∞ -UHF algebra is large in C(X) ⋊α Z. This
is to be expected, though, since C(X) ⋊α Z is in fact the 2∞ Bunce-Deddens alQ
gebra. More generally, any Bunce-Deddens algebra of type i pni i contains (many)
Q
subalgebras isomorphic to the i pni i -UHF algebra which capture the K-theory and
other essential properties of the entire algebra (see, e.g. [9, Section 2], [5, Example
II.10.4.12.ii], [12, Theorem V.3.5]). In fact, a major reason why Bunce-Deddens algebras are so important is because their K-theory can be well-modeled by that of
UHF algebras. This is reinforced by the fact that Bunce-Deddens algebras contain
large UHF subalgebras, since large subalgebras are have almost identical K-theory
and trace structure as the entire algebra in general.
Theorem 3.23. [40] If B ⊂ A is a large subalgebra, then:
1. The inclusion B → A induces an isomorphism of K0 -groups K0 (B) → K0 (A).
2. There exists an exact sequence of K1 -groups 0 → K1 (B) → K1 (A) → Z → 0.
3. The inclusion B → A induces an order isomorphism of Cuntz semigroups
Cu(B) → Cu(A).
4. The radii of comparison are equal, rc(B) = rc(A).
5. B is Z-stable if and only if A is Z-stable.
33
CHAPTER 4. Examples of Minimal Dynamical Systems
The primary objective of this chapter is twofold. First, we exhibit topological dynamical systems that are minimal, infinite-dimensional, and have mean dimension zero.
The primary results of this dissertation are in large part generalizations of existing
results about finite-dimensional dynamical systems to dynamical systems with finite
mean dimension. Theorem 1.1, for instance, is a verification of the Toms-Winter
conjecture (see [58, Conjecture 0.1]) for mean dimension zero systems, a result which
was established for finite-dimensional systems in [54]. To see that the results in this
dissertation are indeed generalizations from existing results, it is important to show
that there do exist many minimal, mean dimension zero systems which are not finitedimensional. Second, since Theorems 1.2 and 7.9 apply to minimal systems whose
product with a Cantor minimal system are still minimal, it is important to show that
such systems are widespread. In this section, we show that this is indeed the case.
In Section 4.1, we construct examples as inverse limits of simpler dynamical systems. In Section 4.2, we use the skew product construction to build more complex
systems as well as prove that minimal, infinite dimensional, mean dimension zero
systems are plentiful. In Section 4.3, we describe an important class of dynamical
systems known as Giol-Kerr subshifts, first developed in [18]. Section 4.4 describes a
class of examples which are important in Theorem 7.9. Specifically, we show how the
product of Giol-Kerr systems with Cantor odometers are minimal. In Section 4.5, we
show that the product of any connected minimal system with a Cantor odometer is
minimal.
34
4.1
Inverse Limit Systems
The properties of minimality and zero mean dimension are both preserved under
inverse limits (see [2, Theorem 2.3 (R1)] and [33, Proposition 6.11], resp.). Therefore
examples of minimal infinite-dimensional systems with mean dimension zero can be
constructed using inverse limits. As stated in Remark 4.1, it is important to note
that duality would give the corresponding crossed product C ∗ -algebras a direct limit
decomposition. This decomposition makes the investigation (and classification) of the
crossed product algebras associated to the dynamical systems in this section a much
simpler task, and hence the examples in this section are not the most satisfying from
a C ∗ -algebraic perspective. However, we have still included these examples because
of their simplicity, ease of construction, and the intuition they provide.
Theorem 4.1. Let (Xn , αn ) be a sequence of topological dynamical systems with factor
maps Xn+1 → Xn . Set (X, α) = lim(Xn , αn ). If each system (Xn , αn ) is minimal and
←−
each mdim(Xn , αn ) = 0, then (X, α) is minimal and mdim(X, α) = 0.
This Theorem makes it easy to construct examples of minimal, infinite-dimensional
systems with mean dimension zero as the inverse limit of minimal, finite-dimensional
systems. Because all such examples are inherently similar, we only provide one:
Example 4.1. Recall from Example B.1.1 that, given an irrational number θ ∈ (0, 1),
the rotation map ρ : T → T given by ρ(z) = e2πiθ z defines a minimal dynamical system
on the circle. Furthermore, recall that if θ, θ′ are two rationally independent irrational
numbers (that is, θ, θ′ do not belong to the same coset in the group R/Q) then the
′
product system (T × T, ρ × ρ′ ) is also minimal (where ρ′ is given by z 7→ e2πiθ z).
In fact, any finite product of irrational rotation systems, provided each rotation is
pairwise rationally independent, is minimal. We use these facts to construct our
desired system.
35
Fix a sequence of rationally independent irrational numbers θn , and consider the
corresponding sequence of rotation homeomorphisms ρn on the circle T. Now set
(Xn , αn ) = (Tn , ρ1 × · · · × ρn ). Construct the natural inverse limit sequence
π
π
2
1
X → · · · → X3 −→
X2 −→
X1
where πn : Tn+1 → Tn is the projection onto the first n factors. Since each system
(Xn , αn ) is minimal and has mean dimension zero (by finite-dimensionality), so does
(X, α) = lim(Xn , αn ). But X is clearly infinite-dimensional, being homeomorphic to
←−
Q∞
n=1 T.
Remark 4.1. As stated at the beginning of this section, the inverse limit construc-
tion does not produce very satisfying examples, especially inverse limits of finitedimensional systems. From a C ∗ -algebra perspective, crossed products coming from
minimal uniquely ergodic finite-dimensional systems are completely classified (see
[55]). Using powerful and well-known tools developed for the classification program,
direct limits of these crossed products are also straightforward to study. Hence the
inverse limits in this section essentially produce ‘nothing new’ from a classification
point of view.
Nevertheless, this perspective contains an interesting idea. It is not unreasonable
to think of minimal mean-dimension zero systems as ‘not far’ from finite-dimensional
systems (minimal here is necessary since the identity map on any topological space
gives a mean dimension zero system). In fact, one could wonder whether all minimal
mean dimension zero systems are in fact inverse limits of minimal finite-dimensional
systems. Such a result would give an elegant alternate route to the results in this
dissertation. As shown in [33, Proposition 6.14], mean dimension zero systems are
inverse limits of finite-entropy systems. However, this gives no result relating to minimality. Another approach could be to use [33, Theorem 5.1] to view mean dimension
zero systems as subshift systems, then try and use the blocking techniques in [18] to
construct minimal, finite-dimensional subsystems which ‘exhaust’ the larger system.
Thus far, however, these approaches have not been successful.
36
4.2
Skew Product Systems
In this section, we construct examples of minimal, infinite-dimensional, mean
dimension zero systems as skew products of other systems:
Definition 4.1. Let (X, α) be a dynamical system and Y a compact space. Any
continuous map f : X → Homeo(Y ) induces a skew product system (X × Y, α ⋉ f )
defined by α ⋉ f : (x, y) 7→ (α(x), fx (y)).
Example 4.2.
1. If f : X → Homeo(Y ) is constant, say fx = β, then the corresponding skew
product is just the product system (X × Y, α × β).
2. The Furstenberg map T2 → T2 defined by (z, w) 7→ (e2πiθ z, zw) for fixed irrational number θ ∈ (0, 1) is a skew product of the irrational rotation (T, θ) with
T, defined by the map f : T → Homeo(T) where fz (w) = zw.
3. If Y = G is a compact metrizable group and f : X → G is any map, we get
a skew product (X × G, α ⋉ f ), thinking of f : X → Homeo(G) as translation
homeomorphisms.
Theorem 4.2. (Adapted from [19, Theorems 1& 2]) Suppose (X, α) is minimal
and uniquely ergodic. If G is a compact, path-connected, metrizable subgroup of
Homeo(Y ), then for a generic set of maps f : X → G, the skew product (X ×Y, α⋉f )
is also minimal and uniquely ergodic.
1
Corollary 4.3. Minimal, infinite-dimensional, mean dimension zero systems exist
(that are not constructed as inverse limits of finite-dimensional systems).
Proof. By [34, Theorem 5.4] strictly ergodic systems (in fact, any system with fewer
than 2ℵ0 ergodic measures) has mean dimension zero. Hence if (X, α) is minimal
and uniquely ergodic, and G is any compact path-connected group (e.g. T∞ ), then
1
This involves a little bit of cheating. In [19], the skew product map has a slightly different definition,
−1
◦ fx )(y)) instead of Definition 4.1.
to wit, α ⋉ f : (x, y) 7→ (α(x), (fαx
37
the minimal skew products (X × G, α ⋉ f ) with mean dimension zero are in fact
generic.
This result is still slightly unsatisfying, since it gives no method for constructing
infinite-dimensional, minimal, mean dimension zero systems; it only states that such
systems are in some sense ‘plentiful.’ We can, however, explicitly construct such
systems using a result of Furstenberg:
Lemma 4.4. Adapted from [16, Lemma 2.1] Suppose (X, α) is a uniquely ergodic
system, and Y is a compact metrizable measure space. If G is a (measure-preserving)
subset of Homeo(Y, ν) which is transitive2 on Y and f : X → Homeo(Y ) is a continuous map with [im f, G] = 1, then the skew product (X × Y, α ⋊ f ) is uniquely
ergodic.
Example 4.3. Let (X, α) be the system from Example 4.1, where X = T∞ and
α(zn ) = (e2πidθn zn ). Note that (X, α) is uniquely ergodic since it is a limit of uniquely
ergodic systems (see [27, Proposition 3.5]). Let Y = T∞ and G = Y ⊂ Homeo(Y ),
viewed as a group acting on itself by translation (obviously, G acts transitively on
Y ). Finally, map f : X → G by the identity. Clearly [im f, G] = 1 since G is abelian.
Therefore (X × Y, α ⋊ f ) is uniquely ergodic (and thus has mean dimension zero as
shown in [34]).
In the above case, the skew product of a mean dimension zero system with a group
action gives another mean dimension zero system, begging the question of how this
relationship can be generalized:
Lemma 4.5. Suppose (X, α) is a dynamical system and Y is a compact metrizable
space. Then for any continuous map f : X → Homeo(Y ), the corresponding skew
product satisfies mdim(X, α) ≤ mdim(X × Y, α ⋊ f ). Furthermore, if f is constant
(say fx = β) and mdim(Y, β) = 0, then in fact mdim(X × Y, α × β) = mdim(X, α).
2
In particular if Y is a group with Haar measure acting on itself by translation.
38
Proof. First, fix metrics d and δ on X and Y respectively, and define the metric
d × δ on X × Y by (d × δ) ((x0 , y0 ), (x1 , y1 )) = max{d(x0 , x1 ), δ(y0 , y1 )}. Next, fixing any y0 ∈ Y , define ϕ : X → X × Y by x 7→ (x, y0 ). By Remark 2.2, recall
that mdim(X, α) = limǫ→0 limn→∞
1
n
Widimǫ (X, dn ). Notice that any ǫ-embedding
g : (X × Y, (d × δ)n ) → K induces a map g ◦ ϕ : X → K. To see that g ◦ ϕ is an
ǫ-embedding, it suffices to show that dn (x0 , x1 ) ≤ (d × δ)n (ϕ(x0 ), ϕ(x1 )), as follows:
dn (x0 , x1 ) = max {d(αk (x0 ), αk (x1 ))}
0≤k≤n
≤ max max{d(αk (x0 ), αk (x1 )), δ(fαk−1 (x0 ) · · · fx0 (y0 ), fαk−1 (x1 ) · · · fx1 (y0 ))}
0≤k≤n
= max (d × δ)((α ⋊ f )k (x0 , y0 ), (α ⋊ f )k (x1 , y0 ))
0≤k≤n
= (d × δ)n (ϕ(x0 ), ϕ(x1 )).
Hence we have an ǫ-embedding (X, dn ) → K.
Thus Widimǫ (X, dn )
≤
Widimǫ (X × Y, (d × δ)n ) and so mdim(X, α) ≤ mdim(X × Y, α ⋊ f ).
Now, suppose we have fx = β for all x ∈ X (so that α ⋊ f = α × β) and
mdim(Y, β) = 0. Then by [34, Proposition 2.8] and the first part of this Lemma, we
have mdim(X, α) ≤ mdim(X × Y, α × β) ≤ mdim(X, α) + mdim(Y, β) = mdim(X, α).
Conjecture 3. Suppose (X, α) is any dynamical system and f : X → Homeo(Y ) is
a continuous map. Denote by G the subgroup of Homeo(Y ) generated by im f . Then
the corresponding skew product satisfies
mdim(X × Y, α ⋊ f ) ≤ mdim(X, α) + mdim(Y, G).
Although skew products present an interesting source of infinite-dimensional systems with mean dimension zero, they can often be difficult to visualize. Indeed, any
topological dynamical system with a factor is isomorphic to a skew product with
that factor as the base (see [1]). For this reason, we restrict our attention to standard products of dynamical systems, and give sufficient conditions for the product of
minimal systems to be minimal.
39
4.3
Giol-Kerr Systems
Giol-Kerr systems were first constructed in [18] to explicitly exhibit a connection
between the mean dimension and the radius of comparison. To date, the systems
constructed in [18] are the only known minimal systems giving a crossed product
with positive radius of comparison (although many are suspected to exist). For this
reason, the Giol-Kerr systems are an important class of examples. In this section,
we define these systems and show that their associated crossed product C ∗ -algebras
satisfy powerful structure theorems.
Definition 4.2. As described in Example B.1.2, any finite simplicial complex K
Q
defines a full shift system (K Z , σ). A block of length ℓ is a closed subset B ⊂ ℓn=1 K.
A block defines a shift-invariant subset (subshift) XB ⊂ K Z , consisting of all points
which can be ‘blocked off’ by B (modulo phase). More precisely, we say a point (xn ) ∈
K Z can be blocked off by B with phase 0 ≤ p ≤ ℓ−1 if (xp+iℓ+1 , xp+iℓ+2 , . . . , xp+iℓ+ℓ ) ∈
B for all i ∈ Z, and denote the set of such points XB,p . Then XB is the set of all
points that can be blocked off by any phase, i.e., XB = ∪ℓ−1
p=0 XB,p .
Example 4.4. Let K = [0, 1] (considering subshifts of the Hilbert cube [0, 1]Z ).
1. Let B = [0, 1], a block of length 1. There is only one possible phase (p = 0),
and every point x ∈ [0, 1]Z is blocked off by B, so XB,0 = XB = [0, 1]Z .
2. Let B = {0} × [0, 1], a block of length 2. Any point x ∈ XB blocked off
by B is of the form (. . . , 0, xi−2 , 0, xi , 0, xi+2 , . . . ). Now there are two phases,
p = 0 and p = 1, corresponding to the zeros being in the even or odd indices.
In general the XB,p are not necessarily disjoint, as shown in this case, where
XB,0 ∩ XB,1 = (. . . , 0, 0, 0, . . . ).
3. Let B be a block of length ℓ, and let B ′ ⊂ B be a sub-block. Then clearly
XB ′ is a subshift of XB . What is more, the block B × B ′ of length 2ℓ defines a
subshift of XB , since any point x ∈ XB×B ′ can also be blocked off by B.
40
Definition 4.3. [18, Section 2] Let K be a finite simplicial complex, and let Bn be a
series of blocks in K Z with lengths ℓ1 ≤ ℓ2 ≤ · · · . Then the intersection X = ∩n XBn
is a subshift of (K Z , σ). We say that (X, σ) is a Giol-Kerr subshift if it satisfies the
following conditions:
1. B1 = K, so XB1 = K Z (this is a non-triviality condition, so that (X, σ) is not
a single point).
2. Each Bn+1 is an extension of Bn (that is, Bn+1 = Bn × B ′ where B ′ is a block
of length ℓn+1 − ℓn ) so that XBn ⊂ XBn+1 .
3. In every block Bn = Bn,1 × · · · × Bn,ℓn , each factor Bn,i is either all of K or a
single point.
4. the distinct phases XBn ,0 , . . . , XBn ,ℓn −1 are pairwise disjoint, and
5. (X, σ) = ∩n XBn is minimal.
Remark 4.2. Let (X, σ) = ∩n (XBn , σ) be a Giol-Kerr system. As above, let ℓn
denote the length of the block Bn . Recall that each Bn is a product of copies of K
and singleton sets, so let dn denote the number of factors of Bn equal to K. Set
θ = lim supn
dn
ℓn
(i.e., θ gives the ‘density’ of K in X). As shown in Corollary 4.9,
mdim(X, σ) = θ dim K. In [18], an algorithm for constructing minimal subshifts
mdim(X, σ) ⊂ ([0, 1]Z , σ) with arbitrary density 0 < θ < 1 was introduced. The
blocks are created so that:
1. Enough factors of Bn = Bn,1 × · · · × Bn,ℓn are single points so that for any two
x, y ∈ XBn , there exists m ∈ N with d(σ m (x), y) < n1 .
2. The remaining factors of Bn are all the full interval [0, 1], with enough so that
the proportion of ‘full’ factors in the block is approximately θ.
Condition (1) ensures that X = ∩n XBn will be minimal, while condition (2) ensures
that the mean dimension is θ dim K.
41
The blocking construction gives the Giol-Kerr algebras nice structural properties
in the form direct limit decompositions.
Lemma 4.6. [18, Lemma 2.1] Let (X1 , α) be a topological dynamical system and let
X2 ⊃ X3 ⊃ · · · be closed α-invariant subsets of X1 . Set X = ∩n Xn . Let
C(X1 ) ⋊ Z → C(X2 ) ⋊ Z → C(X3 ) ⋊ Z → · · ·
be the inductive system with connecting maps induced by the (equivariant) inclusion
maps Xn+1 → Xn , and let ϕn : C(Xn ) × Z → C(X) ⋊ Z denote the maps induced by
X → Xn . Then the direct limit map ϕ : limn C(Xn ) ⋊ Z → C(X) ⋊ Z is an isomor−→
phism.
We will need a slightly stronger version of Lemma 4.6 which proves an analogous
result for Putnam subalgebras (see Definition 3.4). Let (X1 , α) be a topological
dynamical system and let X2 ⊃ X3 ⊃ · · · be closed α-invariant subsets of X1 . Set X =
∩n Xn . Let Y1 ⊂ X1 be a closed subset, and set Yn = Y1 ∩ Xn with Y = ∩n Yn . Notice
that the inclusion Xn+1 → Xn restricts to an inclusion Yn+1 → Yn , and hence induces
a map C(Xn ) ⋊ Z → C(Xn+1 ) ⋊ Z which restricts to a map of Putnam subalgebras
(C(Xn ) ⋊ Z)Yn → (C(Xn+1 ) ⋊ Z)Yn+1 . Denoting by un the unitary in C(Xn ) ⋊ Z
implementing α |Xn and u the unitary in C(X) ⋊ Z implementing α |X (see Definition
3.4), define the map ϕn : (C(Xn ) ⋊ Z)Yn → (C(X) ⋊ Z)Y on generators f ∈ C(Xn )
and un g ∈ un C0 (Xn \ Yn ) by f 7→ f |X and un g 7→ ug |X\Y
Lemma 4.7. The maps ϕn induce an isomorphism
ϕ : lim (C(Xn ) ⋊ Z)Yn → (C(X) ⋊ Z)Y .
−→
Proof. Denote
by
Bn
⊂
C(Xn ) ⋊ Z
the
Putnam
subalgebra
C ∗ (C(Xn ), un C0 (Xn \ Yn )), and denote by AY ⊂ C(X) ⋊ Z the Putnam subalgebra
C ∗ (C(X), uC0 (X \ Y )).
is injective.
By Lemma 4.6, the map lim C(Xn ) ⋊ Z → C(X) ⋊ Z
−→
Since the maps ϕn : Bn → AY are restrictions of the maps
C(Xn ) ⋊ Z → C(X) ⋊ Z defined in Lemma 4.6, it follows that the map lim Bn → AY
−→
42
is a restriction of the isomorphism lim C(Xn ) ⋊ Z → C(X) ⋊ Z. Hence ϕ must
−→
be injective. Furthermore, each ϕn : Bn → AY is surjective (a straightforward
application of the Tietze extension theorem) and hence ϕ must also be surjective.
The most important fact about the Giol-Kerr systems (which makes it possible to
give an upper bound on the radius of comparison) is that approximately mdim(X)proportion of the factors in X are ‘independent.’ The converse of this idea (that is, a
subshift where approximately 0 < θ < 1 proportion of the factors are ‘free’ has mean
dimension θ) is shown in [34]:
Lemma 4.8. [34, Proposition 3.3] Let (X, σ) be a subsystem of the full shift on
(K Z , σ). Suppose that there is an infinite set of indices I ⊂ N and an x̂ ∈ X such
that
1. I has upper density θ, that is, θ = lim supn
1
n
|I ∩ {0, 1, . . . , n − 1}|,
2. any x ∈ K Z with πZ\I (x) = πZ\I (x̂) is in X;
then mdim(X, α) ≥ θ dim(K).
Corollary 4.9. Let (X, σ) = (∩n XBn , σ) be a Giol-Kerr subshift of K Z . Let dn denote
the number of coordinates in the block Bn that are equal to K, and let ℓn denote the
length of Bn . Set θ = lim supn
dn
.
ℓn
Then mdim(X, σ) = θ dim K.
Proof. Notice that (XBn , σ ℓn ) is isomorphic to the full shift on Bn Z (this is ensured
by condition (iv) of Definition 4.3). Since dim(Bn ) = dn dim K, we know that
mdim(XBn , σ) =
1
ℓn
mdim(XBn , σ ℓn ) (by Example 2.1 (2)), and mdim(XBn , σ ℓn ) =
dim(Bn ) (by Example 2.1 (1)). Hence mdim(XBn , σ) = ( dℓnn ) dim K.
By construction, (X, σ) is a closed subsystem of each (XBn , σ). Hence mdim(X, σ) ≤
mdim(XBn , σ) for all n, and so we have the upper bound mdim(X, σ) ≤ ( dℓnn ) dim K
for all n.
The lower bound is a straightforward application of Lemma 4.8. Recall that X
is defined using a blocking procedure, where the coordinates of each block are either
43
single points or all of K. Broadly speaking, we want I to be the list of all indices
that are single points. More specifically, we define a list of indices I ⊂ N inductively
as follows.
Recall that B1 = K, so I1 = ∅. Next, we know B2 = K × B1′ , so let I2 be the set of
all coordinates in B2 consisting of single points (if any). Notice that I2 ⊂ {1, 2, . . . , ℓ2 }
has density
ℓ2 −d2
.
ℓ2
Continuing in the same way, we let In be the list of coordinates in
′
Bn consisting of single points. We will always have In−1 ⊂ In since Bn = Bn−1 ×Bn−1
.
Furthermore, In ⊂ {1, 2, . . . , ℓn } has density
ℓn −dn
.
ℓn
Therefore, defining I = ∪n In , we
know by assumption that the density of I in N is lim supn
ℓn −dn
ℓn
= 1 − θ.
Now, let x̂ ∈ X be any point. By construction, we know that any positive coordinates in X not in I must be equal to all of K. Hence any point x ∈ K Z with πZ\I (x) =
πZ\I (x̂) must be contained in X. So by Lemma 4.8, mdim(X, σ) ≥ θ dim(K).
4.4
Disjoint systems
In this section we construct other minimal, infinite-dimensional, mean dimension
zero systems using the theory of disjointness of dynamical systems. Recall from
Definition B.8 that two minimal systems (X, α) and (Y, β) are disjoint if and only
if their product (X × Y, α × β) is minimal. In [17, Theorem II.3] it was shown that
minimal, distal (see Definition B.9) systems are disjoint from topologically weakly
mixing systems (see Definition B.10). Here we prove Lemma 4.10, which states that
Giol-Kerr systems are weakly mixing (see Definition 4.3). Therefore, their product
with any minimal distal system (in particular, any Cantor odometer) is minimal. This
gives a very important class of minimal, infinite-dimensional systems with arbitrary
mean dimension.
Lemma 4.10. Let (X, σ) be a minimal system of Giol-Kerr type. Then (X, σ) is
weakly mixing.
44
Proof. It is shown in [17] that a system (X, α) is (topologically) weakly mixing if and
only if for any point x ∈ X and open set U ⊂ X the set {n ∈ N | αn (x) ∈ U } is
infinite (also known as {thick sets}-transitive).
Fix a point x ∈ X, an open set U ⊂ X, and consider any y ∈ U . Due to
the specific construction of the Giol-Kerr systems (the blocks contain arbitrary long
strings of singleton factors, and these strings appear infinitely many times) there
exist infinitely many n, ℓ ∈ N such that (xn+1 , . . . , xn+ℓ ) = (yn+i+1 , . . . , yn+i+ℓ ) for
some i ∈ Z. Furthermore, for large enough ℓ, the set U ′ = {z ∈ X | ∃i, n ∈
Z s.t. (zn+1 , . . . , zn+ℓ ) = (yn+i+1 , . . . , yn+i+ℓ )} is an open subset of U . Since αn (x) ∈
U ′ for infinitely many n, (X, σ) is weakly mixing.
Corollary 4.11. If (X, σ) is a Giol-Kerr subshift and (Y, β) is any minimal distal
system (in particular, a Cantor odometer) then (X × Y, σ × β) is minimal.
4.5
Connected systems
As our final source of examples of infinite-dimensional minimal dynamical systems,
we show that the product of any connected minimal system with a Cantor odometer
is minimal. The proof of this fact begins with the following technical Lemma.
Lemma 4.12. [41] Let Y =
Q∞
Fj be a product of finite sets (in particular, Y
Q
Q
could be a Cantor set). For every n ∈ N define Y≤n = nj=1 Fj and Y>n = ∞
j=n+1 Fj
j=1
so that Y = Y≤n × Y>n for each n. Suppose β : Y → Y is a homeomorphism with
the property that for any n ∈ N, there is a transitive permutation σ of Y≤n such that
β({y} × Y>n ) = {σ(y)} × Y>n for every y ∈ Y≤n . Then for any compact metric space
X and any homeomorphism α of X such that all nonzero powers of α are minimal,
the homeomorphism α × β of X × Y is minimal.
m
Proof. We show that for any nonempty open U ⊂ X × Y , ∪∞
m=0 (α × β) (U ) = X × Y .
Of course, it suffices to do this for any subset of U in place of U . For large enough n,
there exists y ∈ Yn and a nonempty open set V ⊂ X such that V × ({y} × Y>n ) ⊂ U .
45
Let N be the cardinality of Yn . Then by transitivity of σ, we know that σ N = idYn
−1
and {σ i (y)}N
i=0 = YN for any y ∈ Yn .
mN +i
Since αN is minimal, for i = 0, . . . , N − 1 we have ∪∞
(V ) =
m=0 α
mN
∪∞
(αi (V )) = X. Now
m=0 α
N −1 mN +i
ℓ
∞
(V ) × β mN +i ({y} × Y>n )
∪∞
ℓ=0 (α × β) (V × ({y} × Y>n )) = ∪m=0 ∪i=0 α
N −1 mN +i
= ∪∞
(V ) × ({σ i (y)} × Y>n )
m=0 ∪i=0 α
−1
mN +i
i
= ∪∞
(V ) × ∪N
m=0 α
i=1 ({σ (y)} × Y>n )
= X × (Y≤n × Y>n )
=X ×Y
as desired.
Example 4.5. As an example of a system satisfying the conditions of Lemma 4.12
let (Y, β) be the 2-odometer, that is, Y = lim Z/2n with β(y) = y + 1 (see Example
←−
Q∞
B.4 for more detail). Notice that as spaces, Y ∼
= j=1 {0, 1} and Y≤n ∼
= Z/2n . In
particular, we can define σ : Z/2n → Z/2n by σ(y) = y + 1 (mod 2n ), which is clearly
a transitive permutation (i.e., satisfies the conditions of the Lemma) since 1 generates
Z/2n .
More generally, if (Y, β) is any odometer, say Y = limj Z/pj , we have that Y≤n ∼
=
←−
Z/pn . Hence we can similarly define σ : Y≤n → Y≤n by σ(y) = y + 1 to satisfy the
desired condition.
3
The second part of Lemma 4.12 requires a system (X, α) such that all nonzero
powers of α are minimal. Although this may seem like an extreme strengthening on
minimality, we next show that it holds for connected systems.
Lemma 4.13. [41] If X is a compact connected space and α : X → X is a minimal
homeomorphism then all nonzero powers of α are minimal.
3
It seems reasonable to conjecture that the ‘transitive permutation’ condition in Lemma 4.12 is
equivalent to (Y, β) being an odometer. Surely, all odometers possess this property. Conversely, the
transitive permutation condition feels almost the same as requiring that (Y, β) be an inverse limit
of finite systems. Since odometers are constructed as inverse limits of finite groups, this conjecture
is not completely unmerited.
46
Proof. Suppose αn is not minimal for some n > 1. Let K ⊂ X be a minimal closed αn invariant set (which exists by a straightforward application of Zorn’s Lemma). Notice
that K ∪ α(K) ∪ · · · ∪ αn−1 (K) is α-invariant, hence K ∪ α(K) ∪ · · · ∪ αn−1 (K) = X.
Furthermore, the αi (K) are pairwise disjoint: for any 1 < i < n, K ∩ αi (K) is αn invariant and hence K ∩ αi (K) = ∅ by minimality of K. Therefore αi (K) ∩ αj (K) =
αi (K ∩ αj−i (K)) = ∅ for any i 6= j. This contradicts the connectedness of X.
Corollary 4.14. If (X, α) is any minimal, connected dynamical system and (Y, β) is
an odometer, then the product system (X × Y, α × β) is minimal.
Proof. Follows directly from Lemmas 4.12 and 4.13.
Example 4.6. Let (X, α) be any minimal, connected, infinite-dimensional system
(e.g. Example 4.1). This Corollary tells us that we can ‘tack on’ a Cantor odometer
without destroying minimality or increasing the mean dimension.
47
CHAPTER 5. Mean Dimension and Z-Stability
In this chapter we prove Theorem 1.1:
Theorem. If (X, α) is a minimal system with mean dimension zero, then
rc(C(X) ⋊α Z) has strict comparison if and only if C(X) ⋊α Z is Z-stable.
Let (X, α) be a minimal system with mean dimension zero. Throughout this
section, we define A = C(X) ⋊α Z. It was shown in [46, Corollary 4.6] that Zstability implies strict comparison for exact, simple, unital C ∗ -algebras (in particular,
for minimal Z-actions). So we need only show the converse. In fact, by Theorem 3.23
it suffices to show that this holds for a large subalgebra (see Definition 3.5). As shown
in Section 3.4.2, A has many large subalgebras of the form A{x} , and each A{x} has
an approximate RSH decomposition (see Definition 3.3) into algebras of the form
AK = · · · Mn1 C(K1 ) ⊕Mn2 C(Z2 ) Mn2 C(K2 ) ⊕Mn3 C(Z3 ) · · · ⊕Mnℓ C(Zℓ ) Mnℓ C(Kℓ )
(5.1)
where Zi = ∂Ki ∩ (K1 ∪ · · · ∪ Ki−1 ).
The algebra AK is naturally included in the subhomogeneous algebra
BK = Mn1 C(K1 ) ⊕ Mn2 C(K2 ) ⊕ · · · ⊕ Mnℓ C(Kℓ )
via a canonical embedding AK ֒→ BK defined in [30, Theorem 3]. The algebra BK
is much more tractable, and hence it is easier to compare elements in Cu(BK ) than
in Cu(AK ). However, the embedding AK → BK is gotten by ‘ignoring’ the patching
that is performed in Equation (5.1) and hence elements in BK cannot be easily lifted
to AK . This is described precisely by the following Lemma:
48
Lemma 5.1. [31, Theorem 3.2] An element b1 ⊕ · · · ⊕ bℓ ∈ BK is contained in AK
if and only if for every x ∈ ∂Ki ∩ Kj1 ∩ αnj1 (Kj2 ) ∩ · · · ∩ α−nj1 −···−njs−1 (Ks ) with
nj1 + nj2 + · · · njs = ni , bi (x) is given by the block-diagonal matrix


b (x)
 j1





bj2 (α−nj1 (x))




−nj1 −nj2
bi (x) = 

bj3 (α
(x))




...




−nj1 −nj2 −···−njs−1
bjs (α
(x)).
Observation 5.1. Recall that (as described in Theorem 3.17) K partitions X into
i −1
Rokhlin towers: X = ⊔ℓi=1 ⊔nj=0
αj (Ki ). However, the ‘bases’ K1 , K2 , . . . , Kℓ of the
towers are not closed. Instead it is their closures that appear in the definition of AK ,
and hence the boundary overlaps must be accounted for. The set described in Lemma
5.1 is the overlap of Ki with the other Rokhlin towers.
If we assume that (X, α) has mean dimension zero (Definition 2.3), then we can
show that the total overlap of the closures of the Rokhlin towers is somehow ‘negligible’. Specifically, defining E = ∪ℓi=1 ∂Ki and F = ∪ℓi=1 αni (E) (notice, in particular,
that F contains the region over which the direct summands in the RSH decomposition
are bonded) then we can show that this bonding region is small (as in Definition 2.4).
Lemma 5.2. If K has small boundary, then F is small. In particular, Remark 5.1
shows that if mdim(X, α) = 0 then we can shrink an arbitrary K so that F is small.
Proof. It is easy to see that F is closed, since each ∂Ki is, and F is a finite union of
homeomorphic images of these. To see that F is small, it suffices to show that each
αni (E) is small since ocap(·) is subadditive. It follows by definition that ocap(A) =
ocap(α(A)) for any A ⊂ X, and hence we need only show that E is small (which
follows if each ∂Ki is small.) Since any subset of a small set is small, it suffices to
show that ∂Ki ⊂ ∪ij=0 αnj (∂K) ∪ ∂K. Recall the following elementary facts, which
hold for any A, B ⊂ X and any homeomorphism α of X.
49
1. ∂(A ∪ B) ⊂ ∂A ∪ ∂B,
2. ∂(A ∩ B) ⊂ ∂A ∪ ∂B,
3. ∂(A \ B) ⊂ ∂A ∪ ∂B,
4. ∂α(A) = α(∂A).
Hence, we have that
nj
∂Ki = ∂ (K ∩ αni (K)) \ ∪i−1
j=0 α (K)
nj
⊂ ∂(K ∩ αni (K)) ∪ ∂(∪i−1
j=0 α (K))
⊂ ∪ij=0 αnj (∂K) ∪ ∂K.
Remark 5.1. Observe that if mdim X = 0, then X has the small boundary property
by [33, Theorem 6.2]. Hence for any point x ∈ X and any neighborhood U of x, there
exists an open set x ∈ V ⊂ U such that V has small boundary. Setting K = V shows
that if mdim(X, α) = 0 we can always provide for ∂K to be small.
Since the complicated structure of AK is due to the patching in its RSH decomposition, it is natural to consider the ideal IK in AK of functions vanishing on this set.
The idea is that since the patching region is small (as shown above) this ideal should
capture much of the structure of AK , while avoiding the complicated pullback definition. This ‘sandwiches’ AK between two subhomogeneous algebras IK ⊂ AK ⊂ BK ,
providing a useful structural result.
Definition 5.1. Define the ideal IK ⊂ AK as follows (compare to Definition 3.3):
IK = Mn1 C0 (K1 \ F ) ⊕ Mn2 C0 (K2 \ F ) · · · ⊕ Mnℓ C0 (Kℓ \ F ).
(5.2)
Most importantly, notice that F contains each Zi = ∂Ki ∩ (K1 ∪ · · · ∪ Ki−1 ), and
hence elements in IK trivially satisfy the pullback conditions defining AK in Equation
(5.1).
50
Our strategy for proving Theorem 1.1 will be to show that A satisfies the conditions
of Theorem 5.3, and hence is Z-stable. To do this, we first reduce to showing the
conditions hold for a subalgebra AK of A. We then use Lemma 5.4 to show that we
can work in the ideal IK ⊂ AK , and finally use Lemma 5.5 to show that the condition
holds in IK .
Theorem 5.3. [11, Corollary 6.3] Let A be a unital simple separable C ∗ -algebra with
locally finite decomposition rank, strict comparison, and nonempty tracial state space
T (A) (in particular, any minimal Z-action or large subalgebras thereof ). Then A is
Z-stable if and only if it satisfies the following condition:
For each positive a ∈ A ⊗ K such that dτ (a) < ∞ for all tracial states τ ,
and each ǫ > 0, there is a positive b ∈ A ⊗ K such that for all τ ,
|dτ (a) − 2dτ (b)| < ǫ.
It is important to note that Theorem 5.3 relies on an important result of Winter
in [57], which states that if the Cuntz semigroup of a unital simple separable C ∗ algebra with locally finite decomposition rank is isomorphic to the Cuntz semigroup
of its Z-stabilization, then the algebra is in fact Z-stable.
The following Lemma shows that elements in AK can be perturbed to elements
in IK without changing their trace.
Lemma 5.4. If K has small boundary, then for every positive element a ∈ AK there
is a positive element b ∈ IK such that dτ (b) = dτ (a) for all traces τ on A.
Proof. By Lemma 5.2, F is small and closed. Hence F is null with respect to every
invariant measure µ on X as shown in Example 2.2.2. It follows that for any descending chain of open sets U1 ⊃ U2 ⊃ · · · with ∩∞
n=1 Un = F , µ(Un ) → 0. By Urysohn’s
Lemma, we can construct a corresponding sequence of continuous ‘bump functions’
hn : X → [0, 1] such that hn |F = 0 and hn |X\Un = 1. By construction, supp(1 − hn ) =
R
Un and so limk (1 − hn )1/k dµ = µ(Un ) → µ(F ) = 0. Equivalently, since invariant
51
measures on X correspond to traces, dτ (1 − hn ) = limk τ ((1 − hn )1/k ) → 0 for any
trace τ .
Setting bn = hn ahn , we know (by the definition of Cuntz subequivalence) that
bn . a. Therefore, the supremum b = sup bn (which exists by [10, First Theorem,
(i)]) is also Cuntz-subequivalent to a, and hence dτ (b) ≤ dτ (a) for all τ . But we also
have that dτ (bn ) ≥ dτ (a) − dτ (1 − hn ) by [11, Lemma 4.2]. Since dτ (1 − hn ) → 0, we
must have dτ (a) = dτ (b).
To show that the ideal IK satisfies the conditions on Theorem 5.3, we use the
following Lemma. This fact is well known, but we include a short proof:
Lemma 5.5. Let X be a locally compact metrizable space and let n ∈ N. For any
a ∈ Mn C0 (X)+ , there exists b ∈ Mn C0 (X)+ such that
|rank(a(x)) − 2 rank(b(x))| ≤ 1.
Proof. Regarding rank(a) as a function X → N, define the sets U0 , . . . , Un ⊂ X by
Uk = rank(a)−1 ({i ∈ N | i ≥ k}). Note that X = U0 ⊃ U1 ⊃ · · · ⊃ Un , where
rank(a) = n on Un (possibly Un = ∅). Since rank(a) is lower-semicontinuous, the Ui ’s
are open. Define functions fi : X → [0, 1] such that supp(fi ) = Ui (such functions can
be constructed using Urysohn’s Lemma) and define bi = ei,i ⊗ f2i for 1 ≤ i ≤ n/2.
Then setting b = ⊕bi gives the desired element.
Theorem 5.6. [10, proof of Theorem 2], see also [50, Proposition 2.4] Let A = lim An
−→
∗
be an inductive limit of C -algebras with connecting maps ϕn : An → An+1 . For every
[a] ∈ Cu(A) there exists a sequence of elements [an ] ∈ Cu(An ) with [ϕn (an )] ≤ [an+1 ]
such that [a] = supn [an ].
Furthermore, if [bn ] ∈ Cu(An ) is another sequence with [ϕn (bn )] ≤ [bn+1 ] and
supn [bn ] = [b] such that [a] ≤ [b], then for any [a′ ] ≪ [an ] in Cu(An ), there exists
m ≥ n such that [ϕm · · · ϕn (a′ )] ≪ [bm ].
Corollary 5.7. Let A = lim An be an inductive limit of C ∗ -algebras with connecting
−→
maps ϕn : An → An+1 . Suppose each An satisfies the divisibility condition described
in Theorem 5.3, then so does A.
52
Proof. Fix a positive element a ∈ A ⊗ K such that dτ (a) < ∞ for all tracial states
τ ∈ T (A), and fix ǫ > 0. Using Theorem 5.6, find an increasing sequence of elements
[an ] ∈ Cu(An ) such that [a] = supn [an ]. By assumption, there exist positive bn ∈
An ⊗ K such that |dτ (an ) − 2dτ (bn )| < ǫ for all τ ∈ T (A). Set [b] = supn [bn ]. Then
|dτ (a) − 2dτ (b)| ≤ |dτ (a) − dτ (an )| + |dτ (an ) − 2dτ (bn )| + 2|dτ (bn ) − dτ (b)| → ǫ.
Proof of Theorem 1.1. Let a ∈ A{x} ⊗ K be as in Theorem 5.3. Given ǫ > 0, our goal
is to construct the desired b ∈ A{x} ⊗ K. Since A{x} ⊗ K ∼
= ∪n Mn (A{x} ) it suffices
to assume that a ∈ MN (A{x} ) for some N . Now, considering aij ∈ A{x} , if we give
bij ∈ A{x} with |dτ (aij ) − 2dτ (bij )| < ǫ (i.e., construct b ∈ MN (A{x} ) entry-wise) the
result then follows for MN (A{x} ) by replacing ǫ with ǫ/N 2 .
So, let a ∈ A{x} be positive and fix ǫ > 0 as in Theorem 5.3. In fact, since A{x} =
lim AK (as shown in Lemma 3.21) and since each dτ is a (lower semi-continuous)
←−
state, we can assume without loss of generality that a ∈ AK for an arbitrarily small
closed neighborhood K of x with nonempty interior. Next, shrink K (if necessary)
so that its interior has small boundary (this is possible by Remark 5.1) and so that
the return times (Definition B.4) of all points in K are greater than 1/ǫ.
Now, we can invoke Lemma 5.4 to find an element c ∈ IK such that dτ (c) = dτ (a)
for all τ . Then, applying Lemma 5.5 gives an element b ∈ IK with |dτ (c) − 2dτ (b)| =
|dτ (a) − 2dτ (b)| < 1/ mink nk < ǫ, since we have assumed each nk > 1/ǫ from the
start. Hence b is the desired element.
53
CHAPTER 6. Mean Dimension as an Upper Bound on the
Radius of Comparison
In this Chapter we prove Theorem 1.2:
Theorem. If (X, α) is a minimal system and (Y, β) is an odometer such that
(X × Y, α × β) is minimal, then
rc (C(X × Y ) ⋊α×β Z) ≤
1
mdim(X).
2
Fist, notice that many minimal systems (X, α) satisfy this condition (Giol-Kerr
systems, weakly mixing systems, connected systems, etc.) as shown in Chapter 4.
Hence Theorem 1.2 applies in considerable generality.
The proof of this theorem will proceed as follows: first, we show that the Putnam
algebra AX×{y} (see Definition 3.4) is large in A = C(X × Y ) ⋊α×β Z. Second, we
completely describe the RSH algebras AX×Yn (see Definition 3.3) and connecting maps
AX×Yn → AX×Yn+1 in the direct limit AX×{y} = lim AX×Yn (see Lemma 3.21). Finally,
−→
1
we show that each rc(AX×Yn ) ≤ 2 mdim(X, α), which implies the final result.
Lemma 6.1. Let (X × Y, α × β) be a minimal system where X and Y are arbitrary
compact spaces. Then for any finite set F ⊂ Y and any open subset W ⊂ X × Y
there exists a finite collection of open sets {Ui } in X × Y covering X × F and integers
{ni } such that {(α × β)ni (Ui )} are disjoint subsets of W .
Proof. First suppose that F is a singleton, so that X × F = X × {y0 }, and let
W ⊂ X × Y be given. Denote by π1 : X × Y → X and π2 : X × Y → Y the standard
coordinate projections.
Let V ⊂ W be a basic open subset of W . Since X × {y0 } is compact and V
is open, we have a finite collection of first-return times {n1 , . . . , nℓ } of the points in
54
(α × β)n1
E1
X
(α × β)n2
E2
V
W
(α × β)ni
Ei
y0
Y
(α × β)−n1
U1
X
(α × β)−n2
U2
W
V1 V2 Vi
(α × β)−ni
Ui
y0
Y
Figure 6.1.: Illustration of Lemma 6.1.
55
X × {y0 } to V (see Definition B.4). Let the sets E1 , . . . , Eℓ denote the corresponding
return-time partition of X × {y0 }
The sets (α × β)ni (Ei ) are disjoint subsets of V . In fact, π2 (α × β)ni (Ei ) =
π2 (α × β)nj (Ej ) (that is, β ni (y0 ) = β nj (y0 )) if and only if i = j, since otherwise this would imply that y0 is a periodic point of (Y, β), an impossibility since
(Y, β) is minimal. Hence we can define disjoint open neighborhoods V1 , . . . , Vℓ of the
(α × β)n1 (E1 ), . . . , (α × β)nℓ (Eℓ ). Moreover, we can construct the Vi so that each is a
subset of W by first finding disjoint open neighborhoods Vi′ of the β ni (y0 ) in Y , and
then setting Vi = π1 (V ) × Vi′ . Now the sets Ui = (α × β)−ni (Vi ) satisfy the desired
properties.
In the case where F is arbitrary, say F = {y0 , . . . , yN }, find disjoint open subsets
W0 , . . . , WN ⊂ W , and repeat the case of N = 0 but using W = Wk each time.
Lemma 6.2. Suppose (X, α) is any minimal system and (Y, β) is a Cantor minimal
system. Then for any y ∈ Y , the algebra AX×{y} is large in A = C(X × Y ) ⋊α×β Z.
Proof. Let ǫ > 0, a1 , . . . , am ∈ A, v ∈ A+ and w ∈ (AX×{y} )+ be as in Definition 3.5.
P
Since finite sums of the form
fn un (where each fn ∈ C(X × Y ) and u ∈ A is the
unitary implementing α × β) form a dense subfamily of A, we can find c1 , . . . , cm ∈ A
P
n
such that each cj = N
n=−N fj,n u satisfies Definition 3.5 (2) with N fixed over all cj .
Next, our goal is to define g ∈ C(X)+ so that it satisfies properties (4.a), (4.c),
and (4.d) of Definition 3.5. First, by [40, Lemma 3.4] find nonzero w0 ∈ C(X)
with 0 ≤ w0 .AX×{y} w. By Lemma 6.1, there exists a finite open cover {Ui } of
X × {β −N (y), β −N +1 (y), . . . , β N −1 (y)} and integers {ni } such that {(α × β)ni (Ui )}
are disjoint subsets of supp(w0 ). Now define g : X × Y → [0, 1] which is supported
on ∪i Ui and such that g = 1 on X × {β k (y)}k=−N,...,N −1 . Notice that g automatically
−1
k
satisfies Definition 3.5 (1), and 1 − g vanishes on ∪N
k=−N (α × β) (X × {y}). So by
Lemma 3.20, (1 − g)cj , cj (1 − g) ∈ AX×{y} for all j and hence Definition 3.5 (3) is
satisfied. Finally, since we constructed g so that {Ui } is an open cover of supp(g)
and {(α × β)ni (U )} are disjoint subsets of supp(w0 ), it follows by Lemma 3.4 that
g . w0 . w.
56
The following lemma is the ‘dual’ to Lemma B.2, which gives an explicit description of how the nice Rokhlin tower decompositions of minimal systems with odometer
factors correspond to nice inverse limit decompositions of the crossed product.
Lemma 6.3. Let (X, α) be any minimal system and let (Y, β) be an odometer such
that (X × Y, α × β) is minimal. Then for any y ∈ Y , the large subalgebra AX×{y}
of A = C(X × Y ) ⋊α×β Z has a direct limit decomposition AX×{y} = lim AX×Yn
−→
into subhomogeneous algebras. Each AX×Yn ∼
M (C(X × Yn )), where Yn are
= lim
−→ pn
compact-open subspaces of Y and the integers pn are given by the odometer structure
of (Y, β).1 The connecting maps AX×Yn → AX×Yn+1 are given by


ρn (a)




−pn
pn


ρn (u au )




−2p
2p
n
n
a 7→ 

)
ρn (u au




...




−(pn+1 −1)pn
(pn+1 −1)pn
)
au
ρn (u
(6.1)
where ρn : Mpn (C(X × Yn )) → Mpn (C(X × Yn+1 )) is induced by the inclusion
X × Yn+1 ֒→ X × Yn and u ∈ A is the unitary implementing α × β.
Q
Qn
Proof. Suppose Y = ∞
i=1 Z/mi , setting pn =
i=1 mi , and fix y ∈ Y . Denoting
Qn
by πn : Y → i=1 Z/mi the projection onto the first n factors, define the subspaces
Yn ⊂ Y by Yn = πn−1 (πn (y)), that is, Yn = {y ′ ∈ Y | πn (y ′ ) = πn (y)}. Clearly, each
Yn is compact, open, and ∩n Yn = {y}.
In particular, consider Y1 = {y1 } × Z/m2 × Z/m3 × · · · . Notice that β cyclically
permutes Y1 among the disjoint subsets Y1 , β(Y1 ), . . . , β m1 −1 (Y1 ). That is to say Y1
decomposes Y into a single Rokhlin tower of height p1 = m1 (see Definition B.5).
In general, the Rokhlin tower associated to Yn ⊂ Y consists of exactly one tower of
height pn , and hence the subset X × Yn ⊂ X × Y also induces a Rokhlin tower of
height pn .
1
Specifically, the pn are the integers appearing in the inverse limit definition definition of the profinite
group Y = lim Z/pn (see Definition B.7).
←−
57
We
The
maps
know
by
important
Theorem
question
3.17
now
that
is
AX×Yn
determining
Mpn (C(X × Yn )) → Mpn+1 (C(X × Yn+1 ))
C(X × Y ) ֒→ Mpn (C(X × Yn )) by

f|
 X×Yn


(f ◦ α−1 ) |X×Yn

f 7→ 
...


are
∼
=
Mpn (C(X × Yn )).
how
the
defined.
connecting
We
know

(f ◦ α−pn +1 ) |X×Yn
Similarly, uχX×(Y \Yn ) is included in Mpn (C(X × Yn )) as the element


0




1 0



 .. ..

.
. 



1 0



.



(6.2)
Since the map Mpn (C(X × Yn )) → Mpn+1 (C(X × Yn+1 )) is completely determined
by the image of C(X × Y ) and uχX×(Y \Yn ) , it must be defined by


ρ (a)
 n



−p
p
n
n


ρn (u au )




−2pn
2pn
a 7→ 
.
)
ρn (u au




...




ρn (upn+1 −pn au−(pn+1 −pn ) )
Example 6.1. Let (Y, β) be the 2∞ -odometer, so that Y =
Q∞
i=1
Z/2, and fix y =
(0, 0, . . . ) ∈ Y . Then Yn = {(yi ) ∈ Y | y1 = y2 = · · · = yn = 0}. By Lemma 6.3, the
RSH decomposition of AX×{0} is given by
C(X ×Y ) → M2 C(X ×Y1 ) → M4 C(X ×Y2 ) → · · · → M2n C(X ×Yn ) → · · · → AX×{0}
where M2n C(X × Yn ) → M2n+1 C(X × Yn+1 ) is given by


a |X×Yn+1
0
.
a 7→ 
n
−n
0
u a |X×Yn+1 u
58
In particular, given f ∈ C(X × Y ), we can follow its path


f |X×Y1
0

f 7→ 
0
f ◦ α−1 |X×Y1


f|
0
0
0
 X×Y2



 0

f ◦ α−1 |X×Y2
0
0

7→ 


−2
 0

0
f ◦ α |X×Y2
0


−3
0
0
0
f ◦ α |X×Y2
7→ · · · .
The following lemma uses the definition of mean dimension, specifically the idea
of an ǫ-embedding (see Definition 2.2) to give an analogous property for C ∗ -algebras.
Lemma 6.4. Let F ⊂ Mn (C(X)) be any uniformly equicontinuous family (in particular, any finite family) and let ǫ > 0. Then there exists δ > 0 with the property:
Let U be any finite open cover of X with diam(U ) < δ for all U ∈ U, and
denote the corresponding nerve map ηU : X → K(U) (or equivalently let
η : X → K be a δ-embedding into a finite simplicial complex, see Remark
2.2). Then there exists a family {â}a∈F ⊂ Mn C(K(U)) such that each
kηU∗ â − ak < ǫ.
Proof. It suffices to prove the case for n = 1 (the general result following by adjusting
δ) so let F ⊂ C(X). By the definition of uniform equicontinuity, there exists δ > 0
such that d(x, y) < δ implies |f (x) − f (y)| < ǫ for any f ∈ F . In particular, for any
1 R
f
dµ
−
f
(x)
x ∈ X and any neighborhood U of x with diam(U ) < δ, µ(U
< ǫ for
) U
all f ∈ F (where µ can be any Radon measure on X).
Now, let U be any finite open cover of X with diam(U ) < δ for all U ∈ U. Recall
P
that the nerve map ηU : X → K(U) is defined by ηU (x) = U ηU (x)[U ], where {ηU }
is any partition of unity subordinate to U. Now for any f ∈ F , define fˆ: K → C by
P
P tU R
f dµ (it is straightforward to check that fˆ is continuous).
fˆ( U tU [U ]) = U µ(U
) U
59
Finally, to see that kfˆ ◦ ηU − f k < ǫ, notice that for any x ∈ X,
X
ˆ
ˆ
=
(
f
◦
η
)(x)
−
f
(x)
f
(
η
(x)[U
])
−
f
(x)
U
U
U
X η (x) Z
U
f dµ − f (x)
=
µ(U ) U
U
Z
X η (x) Z
X
η
(x)
U
U
f dµ −
f (x) dµ
=
µ(U ) U
µ(U ) U
U
U
X η (x) Z
U
=
(f − f (x)) dµ
µ(U ) U
U
X ηU (x) Z
|f − f (x)| dµ
≤
µ(U ) U
U
X ηU (x)
<
µ(U )ǫ
µ(U )
U
= ǫ.
We can now use Lemma 6.4 to show that the positive elements lifted from finite
simplicial complexes are eventually dense in the overall algebra. Recall the notation
from Lemma 6.3: the integers pn are defined by Y = lim Z/pn , given a finite cover U
←−
−1
−n+1
of (X, α) we set Un = U ∨ α U ∨ · · · ∨ α
U, K(Un ) denotes the nerve of Un , and
ηn : X → K(Un ) the corresponding nerve map.
Corollary 6.5. Let F ⊂ AX×{y} be a finite set and let ǫ > 0. Then there exists N ∈ N such that the image Mpn (C(K(Upn ) × Yn )) → AX×Yn → AX×{y} contains F (up to ǫ) for all n ≥ N .
That is, for all n ≥ N there exists a ∗-
homomorphism ϕn : Mpn (C(K(Upn ) × Yn )) → AX×{y} such that for any a ∈ F there
exists â ∈ Mpn (C(K(Upn ) × Yn )) such that kϕn (â) − ak < ǫ.
Proof. Let F ⊂ AX×{y} , and let ǫ > 0. Since AX×{y} = lim AX×Yn , we can find N ∈ N
−→
and {ã}a∈F ⊂ AX×{y} such that kã − ak < ǫ/2 for all a ∈ F and n ≥ N .
60
By Lemma 6.4, we can find an open cover U for X and a family {â}a∈F such that
k(ηU × i)∗ (â) − ãk < ǫ/2, where (ηU × i)∗ : Mpn (C(K(U) × Yn )) → AX×Yn denotes the
∗-homomorphism induced by the map ηU × i : X × Yn → K(U) × Yn .
Denote by ϕn : Mpn C(K(Upn ) × Yn ) → AX×{y} the composition of (ηpn × i)∗ with
the inclusion AX×Yn → AX×{y} . Since Upn refines U, we have for all a ∈ F that
kϕn (â) − ak ≤ kϕn (â) − ãk + kã − ak < ǫ/2 + ǫ/2.
Proof of Theorem 1.2. Set A = C(X × Y ) ⋊α×β Z where (X, α) is a minimal system
and (Y, β) is an odometer. Our goal is to prove that rc(A) ≤ 21 mdim(X, α).
First, we know by Lemma 6.2 that AX×{y} is large in A for any y ∈ Y . Hence by
Theorem 3.23 it suffices to show that rc(AX×{y} ) ≤
1
2
mdim(X, α). By [53, Lemma
4.3], it suffices to prove that the radius of comparison holds on a dense subset
of M∞ (AX×{y} ). In particular, by Lemma 6.4, it suffices to consider the images
ϕn : M∞ (Mpn C(K(Upn ) × Yn )) → M∞ (AX×{y} ) over all finite open covers U of X.
To this end, let a, b ∈ Mpn (C(K(Upn )×Yn )), and consider their images ϕn (a), ϕn (b) ∈
AX×{y} . Without loss of generality we can assume that a and b are defined over the
same K(Upn ) by taking a common refinement if necessary.
We assume that dτ (ϕn (a))+ 12 mdim(X, α) < dτ (ϕn (b)) for all traces τ ∈ T (AX×{y} ).
But by definition of the map X × {y} → X × Yn → K(Upn × Yn ), this ensures that
rank(a) + 21 mdim(X, α) < rank(b). But
dim(K(Upn ) × Yn ) + 1
2pn
pn mdim(X, α) + 2
≤
2pn
mdim(X, α)
→
.
2
rc(Mpn (C(K(Upn ) × Yn ))) ≤
Hence a . b and so ϕn (a) . ϕn (b).
Remark 6.1. Suppose that a ∗-homomorphism A → B induces an order embedding
Cu(ϕ) : Cu(A) → Cu(B) (i.e., x . y if and only if ϕ(x) . ϕ(y)). Then rc(A) ≤ rc(B).
To see this, suppose that x, y ∈ M∞ (A)+ with dτ (x)+r ≤ dτ (y) for all τ ∈ QT(A) and
x 6. y. Then ϕ(x) 6. ϕ(y) (since ϕ is an order embedding) and for all σ ∈ QT(B),
61
dσ (ϕ(a)) + r = dϕ∗ (σ) (a) + r ≤ dϕ∗ (σ) (b) = dσ (ϕ(b)), where ϕ∗ denotes the map
QT(B) → QT(A) induced by ϕ.
If we can show that the map C(X) ⋊α Z → C(X × Y ) ⋊α×β Z induced by the
factor map X × Y → X is an order embedding, then we could remove the factor of
Y from Theorem 1.2. There is hope that this is not an impossible task. For example,
as shown in [3, Theorem 5.12], if dim Y ≤ 1 and K1 (A) = 0 then there is an order
embedding Cu(C(Y, A)) → Lsc(Y, Cu(A)).
62
CHAPTER 7. Mean Dimension as a Lower Bound on the
Radius of Comparison
In this Chapter, we prove Theorem 1.3:
Theorem. Let (X, σ) be a Giol-Kerr subshift. Then
rc(C(X) ⋊σ Z) ≥
1
mdim(X, σ).
2
As a companion to Theorem 1.2, we also show that rc(C(X × Y ) ⋊σ×β Z) ≥
1
2
mdim(X, σ)
where
(Y, β)
is
a
Cantor
minimal
system,
proving
that
rc(C(X × Y ) ⋊σ×β Z) = 21 mdim(X, σ) in these cases.
If (X, σ) is a Giol-Kerr subshift of (K Z , σ) then mdim(X, σ) = θ dim K (for constant 0 < θ < 1) as shown in Corollary 4.9. In [18, Theorem 2.2] it was shown that
rc(C(X) ⋊σ Z) ≥
1
3
dim K − 1 under the condition that
1
3
dim K − 1 <
θ
3
dim K. In
Theorem 1.3, we expand on the techniques used in the proof of [18, Theorem 2.2] to
show that rc(C(X) ⋊σ Z) ≥ 2θ dim K = 12 mdim(X, σ), a bound we conjecture is tight.
The general strategy for proving Theorem 1.3 will be to construct two positive
elements a, b ∈ M∞ (C(X)⋊σ Z) such that a 6. b and yet dτ (a)+ 21 mdim(X, σ) ≤ dτ (b).
The construction of these positive elements is rather involved and we will need a large
body of supporting results. We proceed as follows: in Section 7.1 we review techniques
developed by J. Villadsen and A. Toms for constructing positive elements bounding
the radius of comparison of a commutative C ∗ -algebra. In Section 7.2 we review
Giol and Kerr’s direct limit decomposition of C(X) ⋊σ Z. Additionally, we prove
several technical lemmas that will be used to extend the results from Section 7.1 from
commutative C ∗ -algebras to the crossed product. Primarily, we construct a diagram
as shown in Figure 7.1, compute the maps explicitly, and show that it commutes.
Mℓn (C(Bn ))
λn
/ Mℓ
n+1
(C(Bn+1 ))
C(X1 ) ⋊σ Z
/ ···
ϕn−1
/ C(Xn ) ⋊σ Z
ϕn
/ C(Xn+1 ) ⋊σ Z
Mℓn (C(Bn ))
n+2
(C(Bn+2 ))
Mℓn+1 (C(Bn+1 ))
λn+2
/ ···
ψn+2
ϕn+1
/ C(Xn+2 ) ⋊σ Z
δn+1
δn
/ Mℓ
ψn+1
ψn
ϕ1
λn+1
ϕn+2
/ ···
/ C(X) ⋊σ Z
δn+2
Mℓn+2 (C(Bn+2 ))
Figure 7.1.: Commutative diagram used in the proof of Theorem 1.3.
63
64
Finally, in Section 7.3, we use the results in the previous two sections to construct
the desired positive elements and show that they bound the radius of comparison as
claimed.
7.1
Obstructions in the Cuntz semigroup of a commutative C ∗ -algebra
In this section we briefly summarize the Chern obstruction argument of Villadsen,
which was further developed by Toms and Winter.
Lemma 7.1. [56, Lemma 1] Let X be a compact connected metric space. Let p be
vector bundle over X of dimension n and t1 the trivial rank-one bundle. If [p×k ]−[t1 ] ∈
K0 (X k )+ , then chkn (p×k ) = 0.
Corollary 7.2. [54, Section 4.1] Let X be a compact connected metric space and p
a vector bundle over X. If chn ([p]) 6= 0, then [tm ] 6≤ [p] for any m > rank(p) − n.
Lemma 7.3. Let K be a finite simplicial complex. Then there exists a compact
connected neighborhood retract A ⊂ U ⊂ K of dimension dim A ≥ dim K − 2 and a
vector bundle pA over A such that rank(pA ) = d = 21 dim A and cd (pA ) 6= 0.
Proof. Since the highest-dimensional cell of K will be isomorphic to [0, 1]dim K , it
suffices to assume that K = [0, 1]dim K . Let A ∼
= S n ֒→ [0, 1]dim K where n is the
largest even integer less than dim K. Then the complexification T A ⊗ C of the
tangent bundle T A over A fulfills the desired properties.
Lemma 7.4. [52, Lemma 2.1] Let A be a C ∗ -algebra and let p, q ∈ A be projections.
If there exists x ∈ A with kp − xqx∗ k < 1/2 then p . q.
Corollary 7.5. [52, Lemma 2.1] Let K be a finite simplicial complex. Then there
exist positive elements a, b ∈ M∞ (C(K)) such that:
1. rank(a) = 1,
2. rank(b) ≥ 21 dim K − 1, and
65
3. ka − xbx∗ k ≥ 1/2 for all x ∈ M∞ (C(K)).
Proof. By Lemma 7.3, there exists a connected neighborhood retract A ⊂ U ⊂ K and
a vector bundle p over A with rank(p) = d ≥ 21 dim K − 1 and cd (p) 6= 0. Denoting by
r : U → A the retraction, let b̃ denote the projection in M∞ (C(U )) corresponding to
the pullback r∗ p over U (notice that b̃ |A ∈ M∞ (C(A)) is a projection corresponding to
p). By Urysohn’s Lemma there exists a continuous function f : K → [0, 1] such that
f |A = 1 and f |K\U = 0. Denoting by tk ∈ M∞ (C(K)) the projection corresponding
to the trivial rank-k bundle over K, define the positive elements a = t1 and b =
f (ã) + (1 − f )td .
Clearly rank(a) = 1 and rank(b) ≥
1
2
dim K − 1 by construction. Furthermore,
since b |A = p and chd (p) 6= 0, Lemma 7.1 implies that a |A 6. b |A and hence a 6. b.
Therefore ka − xbx∗ k ≥ 1/2 for all x ∈ M∞ (C(K)) by Lemma 7.4.
7.2
Structure of Giol-Kerr crossed products
Let (X, σ) be a Giol-Kerr system as in Definition 4.3. By definition, X is the
intersection of a descending chain of subshifts (XBn , σ) ⊂ (K Z , σ), where XBn denotes
the set of points blocked off by Bn . For simplicity, we will write Xn for XBn .
As assumed in Definition 4.3, Xn can be decomposed into a cycle of ℓn disjoint
compact subspaces Xn = Yn ⊔ σ(Yn ) ⊔ · · · ⊔ σ ℓn −1 (Yn ), where ℓn denotes the length
ℓ
of Bn and σ ℓn (Yn ) = Yn . Each block Bn+1 is constructed as a sub-block of Bnn+1
/ℓn
,
ensuring that Xn+1 ⊂ Xn and Yn+1 ⊂ Yn .
Remark 7.1. Corollary 3.12 implies that C(Xn ) ⋊σ Z ∼
= Mℓn (C(Yn ) ⋊σℓn Z). But
the subspace Yn corresponds to the ℓn possible phases of points blocked off by Bn ,
and hence Yn ∼
= Bn Z . Thus (Yn , σ ℓn ) is simply the full shift on Yn =
= Bn ℓn Z ∼
BnZ . Therefore by Corollary 3.10, rc(C(Xn ) ⋊ Z) ≥
rc(C(Bn ))
ℓn
≈
dn dim K
2ℓn
and hence
rc(C(Xn ) ⋊ Z) → 12 mdim(X, σ) by Corollary 4.9. However, the radius of comparison
is not continuous with respect to direct limits, and hence this observation does not
prove Theorem 1.3.
66
Definition 7.1. By definition, the block Bn+1 is given by
Bn+1 = Bn × Bn × · · · × Bn × |∗ × ∗ ×
{z· · · × ∗},
|
{z
}
kn+1 times
sn+1 times
where ∗ denotes a point in Bn . Denote by kn+1 the number of factors of Bn+1 equal
to Bn and by sn+1 the number of factors equal to a single point in Bn . Note that
(kn+1 + sn+1 )ℓn = ℓn+1 . For 1 ≤ i ≤ kn+1 + sn+1 , define the map π n,i : Bn+1 → Bn
as the projection to the ith factor in Bn+1 . Of course, π n,i is only surjective for 1 ≤
i ≤ kn+1 . With a slight abuse of notation, define the corresponding ∗-homomorphism
πn,i : C(Bn ) → C(Bn+1 ) induced by π n,i .
Definition
7.2. Define the diagonal (see [56, Section 2]) homomorphism
λn : Mℓn (C(Bn )) → Mℓn+1 (C(Bn+1 )) by
λn (a) = πn,1 (a) ⊕ πn,2 (a) ⊕ · · · ⊕ πn,kn+1 +sn+1 (a).
For N > n, denote by λn,N : Mℓn (C(Bn )) → MℓN (C(BN )) the map λn,N =
λN −1 ◦ · · · ◦ λn+1 ◦ λn .
Definition 7.3. As noted in Remark 7.1, C(Xn ) ⋊σ Z ∼
= Mℓn (C(Yn ) ⋊σℓn Z). Viewing Yn = Bn Z , denote by dn : Bn → Yn the diagonal embedding. The map dn is
equivariant as a map from the action of the trivial group on Bn to the action of ℓn Z
on Yn (by σ ℓn ). Hence dn defines a ∗-homomorphism which we denote
δn : Mℓn (C(Yn ) ⋊σℓn Z) → Mℓn (C(Bn )).
Definition 7.4. Viewing Yn = Bn Z , denote fn : Yn → Bn the projection onto the
zeroth coordinate. Define the ∗-homomorphism
ψn : Mℓn (C(Bn )) → Mℓn (C(Yn ) ⋊σℓn Z)
as the composition Mℓn (C(Bn )) → Mℓn (C(Yn )) → Mℓn (C(Yn ) ⋊σℓn Z) where the first
map is induced by fn and the second is inclusion.
Observation 7.1. Notice that, since fn ◦ dn = idBn , we have δn ◦ ψn = 1Mℓn (C(Bn )) .
67
ℓ
Definition 7.5. Since Bn+1 ⊂ Bnn+1
(ℓ
n+1
fact, the sets {σ iℓn (Yn+1 )}i=0
/ℓn )−1
/ℓn
, it follows that σ ℓn (Yn+1 ) ⊂ Yn .
are disjoint subsets of Yn .
in : ∪i σ iℓn (Yn+1 ) → Yn the inclusion map.
In
Denote by
The connecting maps in the limit
C(X) ⋊σ Z = lim C(Xn ) ⋊σ Z are induced by the inclusion (Xn+1 , σ) ⊂ (Xn , σ).
−→
Denote these maps by
ϕn : Mℓn (C(Yn ) ⋊σℓn Z) → Mℓn+1 (C(Yn+1 ) ⋊σℓn+1 Z).
The map ϕn is defined by
ϕn (a) = ρn (a) ⊕ σ −ℓn (ρn (a)) ⊕ · · · ⊕ σ −ℓn+1 +ℓn (ρn (a)),
where ρn : C(Yn ) ⋊σℓn Z → C(Yn+1 ) ⋊σℓn+1 Z is the restriction map induced by the
(equivariant) inclusion in : Yn+1 → Yn . We abuse notation slightly and write σ i (a) for
ui au−i . For N > n, denote by ϕn,N : Mℓn (C(Yn ) ⋊σℓn Z) → MℓN (C(YN ) ⋊σℓN Z) the
map ϕn,N = ϕN −1 ◦· · ·◦ϕn+1 ◦ϕn and denote by ϕn,∞ : Mℓn (C(Yn )⋊σℓn Z) → C(X)⋊σ Z
the canonical homomorphism from the direct limit.
Lemma 7.6. Each square
λn
Mℓn (C(Bn ))
/ Mℓ
n+1
(C(Bn+1 ))
ψn+1
ψn
Mℓn (C(Yn ))
ϕn
/ Mℓ (C(Yn+1 ))
n+1
commutes.
Proof. Notice that the square
i ◦σ −iℓn
Yn+1n
/ Yn
fn+1
fn
Bn+1
π n,i
/ Bn
68
commutes for any i ∈ Z/(kn+1 + sn+1 ). Hence
ϕn (ψn (a)) = ρn (ψn (a)) ⊕ ρn (σ −ℓn (ψn (a))) ⊕ · · · ⊕ ρn (σ −ℓn+1 +ℓn (ψn (a))))
= (fn ◦ in )∗ (a) ⊕ (fn ◦ in ◦ σ −ℓn )∗ (a) ⊕ · · · ⊕ (fn ◦ in ◦ σ −ℓn+1 +ℓn )∗ (a)
= (π n,1 ◦ fn+1 )∗ (a) ⊕ (π n,2 ◦ fn+1 )∗ (a) ⊕ · · · ⊕ (π n,kn+1 +sn+1 ◦ fn+1 )∗ (a)
= ψn+1 (πn,1 (a) ⊕ πn,2 (a) ⊕ · · · ⊕ πn,kn+1 +sn+1 (a))
= ψn+1 (λn (a)).
Corollary 7.7. For any n, m ∈ N, the square
Mℓn (C(Bn ))
λn,n+m
/ Mℓ
n+m
(C(Bn+m ))
O
δn+m
ψn
Mℓn (C(Yn ) ⋊σℓn Z) ϕn,n+m/ Mℓn+m (C(Yn+m ) ⋊σℓn+m Z)
commutes.
Proof. By Lemma 7.6,
δn+m ◦ ϕn,n+m ◦ ψn = λn,n+m = δn+m ◦ ϕn+m ◦ · · · ◦ ϕn+1 ◦ ϕn ◦ ψn
= δn+m ◦ ψn+m−1 ◦ λn+m ◦ · · · ◦ λn+1 ◦ λn
= λn+m ◦ · · · ◦ λn+1 ◦ λn
= λn,n+m .
Lemma 7.8. Let p be a vector bundle over Bn (corresponding to a projection in
M∞ (C(Bn )) which we also denote by p). Then
λn,n+m (p) ∼
= p×(kn+m ···kn+2 kn+1 ) ⊕ trn (m)
where rn (m) is defined recursively by rn (0) = sn and rn (m) = kn+m rn (m − 1) + sn+m
(notice that ℓn+m = kn+m · · · kn+2 kn+1 + rn (m)).
69
Proof. We induct on m.
In general, if p is a vector bundle over a compact space X and π1 , π2 : X × X → X
denote the projection maps onto the first and second factors (resp.)
then
π1∗ (p) ⊕ π2∗ (p) ∼
= p × p.
With this in mind, suppose p is a vector bundle over Bn .
the maps π n,i : Bn+1
Recall that
→ Bn are surjective for 1 ≤ i ≤ kn+1 .
Hence
πn,1 (p) ⊕ πn,2 (p) ⊕ · · · ⊕ πn,kn+1 (p) ∼
= p×kn+1 . Since πn,i : Bn+1 → Bn maps to a single
point for kn+1 + 1 ≤ i ≤ kn+1 + sn+1 and all bundles over a single point space are
trivial, πkn+1 +1 (p) ⊕ πkn+1 +2 (p) ⊕ · · · ⊕ πkn+1 +sn+1 (p) ∼
= tsn , where tn denotes the trivial rank-n bundle. Therefore λn (p) = πn,1 (p) ⊕ · · · ⊕ πn,kn+1 +sn+1 (p) ∼
= p×kn+1 ⊕ tsn ∼
=
p×kn+1 ⊕ trn (0) .
Now suppose that λn,n+m (p) ∼
= p×(kn+m ···kn+2 kn+1 ) ⊕ t(sn+m ···sn+1 sn ) . Denote pm =
p×(kn+m ···kn+2 kn+1 ) ⊕ t(s ···s s ) . Since in general (p×m )×n ∼
= p×mn , we have
n+1 n
n+m
λn,n+m+1 (p) ∼
= λn+m+1 (pm )
n+m+1
∼
⊕ tsn+m+1
= p×k
m
∼
= (p×(kn+m ···kn+2 kn+1 ) ⊕ trn (m) )×kn+m+1 ⊕ tsn+m+1
∼
= p×(kn+m+1 kn+m ···kn+2 kn+1 ) ⊕ trn (m+1)
as desired.
7.3
The proof of Theorem 1.3
Proof of Theorem 1.3. Let n ∈ N be arbitrary. As shown in Lemma 7.5, there exists
a positive element b ∈ M∞ (C(Bn )) that restricts to a projection (vector bundle) q on
some subset A ⊂ Bn such that:
1. b has constant rank rank(b) = rank(q) = r,
2. r ≥ 12 dim(Bn ) − 1 =
kn
2
dim K − 1, and
3. The top Chern class chr (q) 6= 0.
70
Let a ∈ M∞ (C(Bn )) denote the positive element corresponding to the trivial rank-1
vector bundle p = t1 over Bn .
The elements a and b define positive elements ψn (a) and ψn (b) over C(Xn ) ⋊ Z.
For any m ∈ N, denote by am and bm the elements ϕn,n+m (ψn (a)) and ϕn,n+m (ψn (b))
in M∞ (C(Xn+m ) ⋊ Z). Finally, denote by a∞ and b∞ the corresponding elements in
C(X) ⋊σ Z (following am and bm along the canonical connecting maps in the direct
limit as m → ∞).
Denote by An+1 ⊂ Bn+1 the subset gotten by replacing each copy of Bn in
Bn+1 with A. Notice that An+1 is homeomorphic to Akn+1 . More generally, for every m ∈ N, denote by An+m ⊂ Bn+m the subset (homeomorphic Akn+m ···kn+2 kn+1 )
gotten by replacing each copy of Bn+m−1 in Bn+m with An+m−1 .
Denote by
ρn+m : Mℓn+m (C(Bn+m )) → Mℓn+m (C(An+m )) the restriction map induced by the inclusion An+m ⊂ Bn+m .
Since a and b restrict to the vector bundles p and q over A ⊂ Bn , it follows by
Lemma 7.8 that for all m we have:
For
ρn+m (λn,n+m (a)) ∼
= tℓn+m ,
(7.1)
ρn+m (λn,n+m (b)) ∼
= q ×(kn+m ···kn+2 kn+1 ) ⊕ trn (m) .
(7.2)
simplicity,
denote
pm = tℓn+m ∈ Mℓn+m (C(An+m ))
and
qm = q ×(kn+m ···kn+2 kn+1 ) ⊕ trn (m) ∈ Mℓn+m (C(An+m )).
Lemma 7.1 shows that p 6. q. Since ℓn+m = kn+m · · · kn+2 kn+1 + rn (m), it follows by Lemma 7.2 that tℓm+n 6. p×(kn+m ···kn+1 ) ⊕ trn (m) , i.e., pm 6. qm for all m.
Therefore, by Lemma 7.4, kpm − xqm x∗ k ≥ 1/2 for all x ∈ M∞ (C(An+m )) and hence
kδn+m (am ) − xδn+m (bm )x∗ k ≥ 1/2 for all x ∈ M∞ (C(Bn+m )).
Since homomorphisms are contractive (and δn is surjective) kam − xbm x∗ k ≥ 1/2
for all x ∈ M∞ (C(Xn+m ) ⋊ Z). Since am → a∞ and bm → b∞ , this implies that
a∞ 6. b∞ .
71
Mℓn C(Bn )
λn
/ Mℓ
n+1
C(Bn+1 )
C(Xn ) ⋊ Z
ϕn
/ C(Xn+1 ) ⋊ Z
/ Mℓ
n+m
C(Bn+m )
/ ···
ϕn+m−1
/ C(Xn+m ) ⋊ Z
δn+1
Mℓn+m C(Bn+m )
ρn+m
Mℓn+1 C(An+1 )
Mℓn C(A)
λn
/ λn (a), λn (b) _
Mℓn+m C(An+m )
λn+1
/ · · ·λ n+m−1/ λn,n+m (a), λn,n+m (b)
_
ψn+1
ψn
ϕn
/ a1 , b 1 _
a,_ b
λn (a),_ λn (b)
p, q
/ ··· ϕn+m−1
/ am , b m _
p 1 , q1
ϕn+m,∞
/ a∞ , b ∞
δn+m
λn,n+m (a), λn,n+m (b)
_
ρn+1
ρn
ψn+m
ϕn+1
δn+1
δn
/ C(X) ⋊σ Z
δn+m
ρn+1
ρn
ϕn+m,∞
Mℓn+1 C(Bn+1 )
a0 ,_ b0 ϕn+1
Mℓn C(Bn )
λn+m−1
ψn+m
δn
a,_ b / ···
ψn+1
ψn
λn+1
ρn+m
p m , qm
Figure 7.2.: Illustration of Theorem 1.3, summarizing all of the notation used in
the proof. Note that we write C(Xn ) ⋊ Z in place of the (isomorphic) algebra
Mℓn (C(Yn ) ⋊ Z).
72
By construction, we can compute dτ (a) =
rc(C(X) ⋊σ Z) ≥
kn dim K−1
2ℓn
−
1
.
ℓn
and dτ (b) ≥
kn dim K−1
.
2ℓn
Hence
Since n was arbitrary, we have
rc(C(X) ⋊σ Z) ≥ lim sup(
n
7.3.1
1
ℓn
1
1
kn dim K − 1
− ) = mdim(X, σ).
2ℓn
ℓn
2
Further Results
With a very minor modification to the proof of Theorem 1.3, we can show the
following:
Theorem 7.9. If (X, σ) is a Giol-Kerr system and (Y, β) is a minimal odometer,
then
rc(C(X × Y ) ⋊σ×β Z) ≥
1
mdim(X, σ).
2
Proof. Because the proof is almost exactly the same as that of Theorem 1.3, we only
sketch a proof here. The essence of the argument is to build a commutative diagram
similar to Figure 7.3. In this case, it will look like the following:
Mℓn C(Bn × Y )
λn
/ · · ·λn+m−1/ Mℓ
n+m
C(Bn+m × Y )
ψn+m
ψn
C(Xn × Y ) ⋊ Z
ϕn
/ ···
ϕn+m−1
/ C(Xn+m × Y ) ⋊ Z
Mℓn C(Bn × Y )
Mℓn+m C(Bn+m × Y )
ρn+m
ρn
Mℓn C(A × Y )
/ C(X × Y ) ⋊σ×β Z
δn+m
δn
ϕn+m,∞
Mℓn+m C(An+m × Y )
Just as before, we can write C(X × Y ) ⋊σ×β Z = lim C(Xn × Y ) ⋊σ×β Z. In this
−→
∼
case, C(Xn × Y ) ⋊σ×β Z = Mℓn (C(Yn × Y ) ⋊σℓn ×β ℓn Z) where Yn ⊂ Xn is the same.
The maps are all defined similarly:
1. ψn : Mℓn (Bn × Y ) → Mℓn (C(Yn × Y ) ⋊ Z) induced by
f × idY : Yn × Y → Bn × Y ,
73
2. λn : Mℓn (Bn × Y ) → Mℓn+1 (C(Bn+1 × Y )) induced by the maps
π n,i × idY : Bn+1 × Y → Bn × Y ,
3. ϕn : Mℓn (C(Yn × Y ) ⋊ Z) → Mℓn+1 (C(Yn+1 × Y ) ⋊ Z) the map from the direct
limit (gotten by the inclusion Xn+1 ⊂ Xn ),
4. δn : Mℓn (C(Yn × Y ) ⋊ Z) → Mℓn (C(Bn × Y )) induced by
dn × idY : Bn × Y → Yn × Y , and
5. ρn : Mℓn (C(Bn × Y )) → Mℓn (C(An × Y )) the restriction gotten by the inclusion
An × Y ⊂ Bn × Y .
It is easy to show that the same maps commute, and dim(Bn × Y ) = dim Bn .
Hence we can repeat exactly the argument from Theorem 1.3.
Combined with Theorem 1.2, we have the following:
Corollary 7.10. If (X, σ) is a Giol-Kerr system and (Y, β) is a minimal odometer,
then
rc(C(X × Y ) ⋊σ×β Z) =
1
mdim(X, σ).
2
In order to extend the lower bound in Theorem 1.3 to all minimal systems we
would be need (at the very least) a converse to Proposition 4.8, which would take the
following form:
Conjecture 4. If (X, α) is an extension of a minimal system, then it can be embedded
into the full shift on ([0, 1]d )Z such that the following holds:
1. There exists a set of indices I ⊂ N and a point x̂ ∈ X such that any x ∈ ([0, 1]d )Z
with πZ\I (x) = πZ\I (x̂) is in X.
2. The set I ⊂ N has upper density
1
d
mdim(X, α).
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74
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APPENDICES
78
APPENDIX A. C ∗ -Algebras and K-Theory
In this appendix we provide background and basic results about C ∗ -algebras, Ktheory, and the Elliott classification program for reference throughout this text. This
appendix borrows extensively from many standard texts.
Some standard references on C ∗ -algebras are [5, 13, 36, 37].
Some standard references on K-theory are [6, 23, 24, 45, 48].
Some standard references on the classification program are [4, 47, 52].
A.1
C ∗ -algebras
Definition A.1. A Banach algebra A is a C-algebra equipped with a submultiplicative norm (kabk ≤ kakkbk for all a, b ∈ A) under which the algebra is complete (a
Banach space) and all algebraic operations are continuous.
An involution on a Banach algebra A is a map ∗ : A → A (written a 7→ a∗ )
satisfying the conditions (a + b)∗ = a∗ + b∗ , (ab)∗ = b∗ a∗ , (αa)∗ = ᾱa∗ , and ka∗ k = kak
for all a, b ∈ A and α ∈ C.
A C ∗ -algebra A is a Banach algebra with an involution that satisfies the C ∗ -equality
ka∗ ak = kak2 for all a ∈ A.
Example A.1.
1. The simplest example of a C ∗ -algebra is C itself, where the involution map is
defined as complex conjugation.
2. Let H be a Hilbert space, and let B(H) denote the space of all bounded operators
on H. This space carries a natural norm and involution (namely, the adjoint
operation) which make B(H) a C ∗ -algebra. More generally, any norm-closed,
self-adjoint subalgebra of B(H) is a C ∗ -algebra with the induced norm and in-
79
volution (e.g. the algebra K(H) of compact operators, or the Toeplitz algebra).
In fact, as shown in Theorem A.2, every C∗-algebra is a subalgebra of B(H) for
some Hilbert space H.
3. Let X be a compact metrizable space. The algebra C(X) of continuous functions X → C caries a norm (the sup-norm kf k = supx∈X |f (x)|) and involution
f ∗ (x) = f (x) which make it a C ∗ -algebra. More generally, if X is a locally compact Hausdorff space, let C0 (X) denote the set of continuous functions X → C
vanishing at infinity (that is, for each f ∈ C0 (X) and any ǫ > 0, there exists
a compact subset K ⊂ X such that |f (x)| < ǫ for all x ∈ X \ K. Equivalently, if αX denotes the one-point compactification of X, C0 (X) is the space
of all continuous functions f : X → C whose extensions f : αX → C satisfy
f (∞) = 0.) Then C0 (X) is a C ∗ -algebra. By the Theorem A.1 (also known as
the fundamental theorem of C ∗ -algebras) all commutative C ∗ -algebras are of this
form.
4. The algebra Mn (C) of n×n matrices with entries in C (often denoted simply Mn ),
under the usual matrix norm with involution given by the conjugate transpose,
is a C ∗ -algebra. More generally, if A is any C ∗ -algebra then Mn (A) is also a C ∗ algebra. In this case, involution is defined entrywise by (aij )∗ = (a∗ji ). Although
the norm on this algebra is not immediately apparent, we can use Theorem A.2
to regard A ⊂ B(H) in which case we naturally have Mn (A) ⊂ B(H ⊕n ) and can
use the inherited norm.
5. The most important class of C ∗ -algebras studied in this dissertation are the
crossed product algebras coming from topological dynamical systems. We give
an overview of these algebras in Appendix C.
Theorem A.1 (Gelfand Theorem). A C ∗ -algebra A is commutative if and only if
A∼
= C0 (X) for some locally compact Hausdorff space X.
Theorem A.2 (Gelfand-Naimark Theorem). Let A be a C ∗ -algebra. Then there
exists a Hilbert space H and an isometric monomorphism A → B(H). Equivalently,
80
every C ∗ -algebra is isometrically isomorphic to a subalgebra of the bounded operators
on some Hilbert space.
Example A.2. Let X be a locally compact Hausdorff space. Then C0 (X) is isomorphic to the subalgebra of multiplication operators in B(L2 (X)).
Definition A.2. Let A be a C ∗ -algebra. An element p ∈ A is a projection if p∗ =
p2 = p. An element a ∈ A is self-adjoint if a = a∗ and normal if aa∗ = a∗ a. Provided
A is unital, an element u ∈ A is a unitary if u∗ u = uu∗ = 1 and a ∈ A is invertible
if there exists a−1 ∈ A with aa−1 = a−1 a = 1. Denote by U(A) the group of unitary
elements in A and GL(A) the set of invertible elements in A.
Definition A.3. The spectrum of an element a ∈ A as the set σ(a) =
{λ ∈ C | a − λ1 6∈ GL(A)}. Any complex polynomial p(z) defines an element p(a) ∈
A. Since every continuous function f ∈ C0 (σ(a)) is the limit of polynomials, the
element f (a) is well-defined.
Theorem A.3 (Spectral Mapping Theorem). Let A be a unital C ∗ -algebra and a ∈ A
a normal element. Denoting by C ∗ (a) ⊂ A the subalgebra generated by 1 and a, the
map defined by
C0 (σ(a)) → C ∗ (a)
f 7→ f (a)
is an isomorphism.
Definition A.4. A trace on a C ∗ -algebra is a positive linear functional τ : A → C
satisfying the condition τ (ab) = τ (ba) for all a, b ∈ A. A 2-quasitrace is a function
τ : A → C that satisfies the following properties:
1. τ is linear on commutative subalgebras of A,
2. for all a ∈ A, τ (a∗ a) = τ (aa∗ ) and τ (a∗ a) ≥ 0,
3. if a, b ∈ A are self-adjoint, then τ (a + ib) = τ (a) + iτ (b), and
81
4. for all n, τ extends to a map on Mn (A) with the same properties.
We denote by QT2 (A) the set of 2-quasitraces on A.
Theorem A.4. [22] Every 2-quasitrace on an exact unital C ∗ -algebra is a trace.
Remark A.1. Since all of the C ∗ -algebras considered in this dissertation are exact
(in fact, the crossed products we consider are nuclear, a stronger property) we can
conflate quasitraces and traces without risk of confusion.
A.2
K-Theory
In the following sections, we give brief definitions of the K0 and K1 groups of a C ∗ algebra. We also give a brief review of topological K-theory, the Chern character, and
describe connections between topological K-theory and K-theory for C ∗ -algebras.
A.2.1
The K0 -group of a unital C ∗ -algebra
Definition A.5. Let A be a unital C ∗ -algebra and denote by M∞ (A) the algebraic
direct limitA →M2 (A) → M3 (A) → · · · → ∪n Mn (A) = M∞ (A) with connecting
a 0
. Denote by P∞ (A) the set of projections in M∞ (A). We say that
maps a 7→ 
0 0
two projections p, q ∈ M∞ (A) are Murray-von Neumann equivalent, written p ∼MvN q,
if there exists v ∈ M∞ (A) such that p = v ∗ v and q = vv ∗ .
Example A.3. Let p ∈ Mn (C) be a projection. Considering p as a linear operator
p : Cn → Cn , let V ⊂ Cn be the range of p. Now suppose q ∈ Mn (C) is another
projection with range W . If dim(V ) = dim(W ), then there exists a linear isomorphism
v : V → W which extends to an element v ∈ Mn (C). Since v ∗ : W → V is also
an isomorphism, this element satisfies v ∗ v = p and vv ∗ = q (and hence p ∼ q).
Conversely, it is easy to see that if p ∼ q then dim(range(p)) = dim(range(q)).
82
Definition A.6. Denote by V (A) the set of Murray-von Neumann equivalence classes
ofprojections

 in M∞ (A). This set is a semigroup with addition defined by [p] + [q] =
p 0

 and identity element [0].
0 q
Example A.4.
1. In the case where A = C, P∞ (A) consists of projections in M∞ (C). Since two
projections in M∞ (C) are equivalent if and only if they have the same rank,
V (C) ∼
= N.
2. If A = B, then P∞ (B) consists of projections in M∞ (B) ∼
= B. Since two projections in B are equivalent if and only if their ranges are isomorphic (as Hilbert
spaces), V (B) ∼
= N ∪ {∞}.
3. If A = B/K, then V (B/K) ∼
= {0, ∞}.
Definition A.7. If S is an abelian semigroup with identity, the Grothendieck enveloping group G(S) of S is the group of formal differences of elements in S. Explicitly,
define an equivalence relation on S × S by (s1 , s2 ) ∼ (t1 , t2 ) if s1 + t1 + r = s2 + t2 + r
for some r ∈ S (Informally, thinking of (s1 , s2 ) as ‘s1 − s2 ,’ we want s1 − s2 ∼ t1 − t2 .
The element r is necessary since cancellation does not work for general semigroups.)
Then define G = M × M/ ∼ with addition inherited from M × M . The identity
element is [(0, 0)] and −[(s1 , s2 )] = [(s2 , s1 )].
If S has cancellation (that is, s+r = t+r implies s = t) then S is a sub-semigroup
of G(S). If S is in fact a group, then S ∼
= G(S).
Example A.5. The Grothendieck enveloping group of N is Z.
Definition A.8. The K0 -group of a C ∗ -algebra A, denoted K0 (A), is the
Grothendieck enveloping group of the semigroup V (A).
When A is stably finite, the semigroup V (A) induces a partial order on K0 (A).
Defining positive cone K0 (A)+ inside K0 (A) as the image of V (A) in K0 (A) via the
Grothendieck construction, we say that x ≤ y if y − x ∈ K0 (A)+ . Note that the
83
positive cone K0 (A)+ ⊂ K0 (A) is isomorphic to V (A) if and only if A has cancellation
of projections (see e.g. [51]). By Theorem 3.2, A = C(X) has cancellation if dim(X) ≤
3. However, cancellation fails in higher dimensions, as shown in Example 3.2.
Example A.6. The K0 -group of the C ∗ -algebra C is Z, and its positive cone K0 (C)+
is N.
A.2.2
The K0 -group from an A-module perspective
Definition A.9. Let A be a unital C ∗ -algebra. Recall that an A-module E is finitely
generated if there exists a finite set {e1 , . . . , en } whose A-linear span is E. Furthermore, E is projective if E is a direct summand of a free A-module (that is, there exists
an A-module F with E ⊕ F ∼
= An ).
Example A.7.
1. In the case where A = C, finitely generated projective C-modules are simply
vector spaces over C.
2. A is a finitely generated projective module over itself. More generally, the n-fold
direct sum An is a finitely generated projective A-module.
3. If E and F are finitely generated projective A-modules then their direct sum
E ⊕ F is also finitely generated and projective.
4. If p is a projection in Mn (A), then its image im(p) ⊂ An is a finitely generated
projective A-module.
Lemma A.5. [23, Lemma A.4.4] There is a one-to-one correspondence between
Murray-von Neumann equivalence classes of projections in M∞ (A) and isomorphism
classes of finitely-generated projective A-modules given by [p] 7→ [pAn ].
Definition A.10. We say that two A-modules E and F are isomorphic if there is an
A-linear isomorphism E → F , and denote by V (A) the set of isomorphism classes of
finitely generated projective A-modules. The set V (A) is an abelian semigroup with
addition given by [E] + [F ] = [E ⊕ F ] and identity element [0].
84
Example A.8. In the case where A = C, two finitely generated projective C-modules
(that is, vector spaces over C) are isomorphic if and only if they have the same rank
(dimension). Hence V (C) ∼
= N.
Definition A.11. The K0 -group of a unital C ∗ -algebra A, denoted K0 (A), is the
Grothendieck enveloping group of V (A). Under the correspondence defined in Lemma
A.5, this definition of the K0 -group is isomorphic to Definition A.8.
A.2.3
Topological K-theory
Before the K-theory was developed for C ∗ -algebras, it was defined as an extraordinary cohomology theory for algebraic geometry. Although less general than C ∗ algebraic K-theory, topological K-theory provides both an important geometric perspective and useful tools for computing the K-groups of C ∗ -algebras. We give a brief
review here.
Definition A.12. Let X be a locally compact Hausdorff space. A vector bundle over
X is a topological space E and a map p : E → X satisfying the following properties.
1. The map p : E → X is a continuous surjection.
2. For every x ∈ X, the fiber Ex = p−1 (x) has the structure of a finite-dimensional
complex vector space compatible with the topology inherited from E.
3. For every x ∈ X, there is a neighborhood U of x and an isomorphism
ϕ : U × Cn → p−1 (U ) for some positive integer n. Furthermore, this isomorphism satisfies the condition p ◦ ϕ(x, v) = x for all (x, v) ∈ U × Cn .
We often denote by p the vector bundle p : E → X. If X is connected, the third
condition implies that the dimension n of each fiber is the same. We call this the
rank of the vector bundle.
Example A.9.
1. The map X × Cn → X defined by projection onto X is a vector bundle, called
the trivial rank n vector bundle, and is denoted tn : X × Cn → X.
85
2. Let X be a smooth manifold, and denote by Tx the tangent space at x ∈ X.
Defining T X = ⊔x∈X Tx , the map p : T X → X defined on each Tx by p |Tx (v) = x
is a vector bundle over X, called the tangent bundle on X.
3. Let V be a complex vector space and let X be its projective space. Define
the space E ⊂ X × V by E = {(x, v) | v ∈ x}. Then p : E → X defined by
p(x, v) = x is a vector bundle.
4. We say that the vector bundle q : F → X is a sub-bundle of p : E → X if F ⊂ E
and q = p |F .
5. If p : E → X and q : F → X are two vector bundles over X, then their Whitney
sum p ⊕ q : E ⊕ F → X is a vector bundle over X with fiber Ex ⊕ Fx at each
x ∈ X.
6. If p : E → X is a vector bundle and f : Y → X is a continuous map,
we can form the pullback bundle f ∗ p : f ∗ E → Y as follows:
f ∗E =
{(y, e) ∈ Y × E | f (y) = p(e)} and (f ∗ p)(y, e) = y.
7. If p is a vector bundle over X and q is a vector bundle over Y , their Cartesian
product p × q is a vector bundle over X × Y which has fiber p−1 (x) × q −1 (y) at
the point (x, y). If πX : X × Y → X and πY : X × Y → Y denote the projections
∗
onto X and Y then πX
p ⊕ πY∗ q ∼
= p × q. We usually denote the vector bundle
p × · · · × p over X n by p×n .
| {z }
n times
Definition A.13. We say that the vector bundles p : E → X and q : F → X over X
are isomorphic (denoted p ∼
= q) if there exists a homeomorphism ϕ : E → F such that
q ◦ ϕ = p and for all x ∈ X, ϕ |Ex : Ex → Fx is a linear isomorphism. We denote by
Vect(X) the set of isomorphism classes vector bundles over X. We say that p and q
are stably isomorphic if there exist trivial bundles tn and tm such that p ⊕ tn ∼
= q ⊕ tm .
Lemma A.6. The set Vect(X) has the structure of an abelian semigroup with addition given by [p] + [q] = [p ⊕ q] and identity element [t1 ].
86
Definition A.14. The K 0 -group of X, denoted K 0 (X), is defined as the
Grothendieck enveloping group of the semigroup Vect(X). Just like V (A), we can
define a partial order on Vect(X) by [p] ≤ [q] if p is isomorphic to a sub-bundle of q.
This extends to a partial order on K 0 (X).
Example A.10. Let X = S 2 . We define the Bott bundle b over S 2 by first identifying
S2 ∼
= CP1 , the projective space of 1-dimensional (complex) linear subspaces of C2 .
Define Eb = {(x, v) ∈ CP1 × C2 | v ∈ x} and then b : Eb → S 2 by b(x, v) = x. As
it turns out, the Bott bundle is nontrivial. What is more, K 0 (S 2 ) ∼
= Z ⊕ Z and is
generated by the elements [t1 ] and [b] . Put another way, the Bott bundle generates
the reduced K0 -group of S 2 .
Theorem A.7 (Serre-Swan Theorem). If X is a compact Hausdorff space, the category of finitely generated projective C(X)-modules is equivalent to the category of
vector bundles on X. Under this equivalence, K0 (C(X)) ∼
= K 0 (X).
Example A.11. Let X be a compact Hausdorff space.
To a projection p ∈
Mn (C(X)) we can associate the map p′ : Ep → X as follows:
define Ep =
{(x, v) ∈ X × Cn | v ∈ p(x)Cn } and p′ (x, v) = x. Then p′ is a vector bundle and
the isomorphism in Theorem A.7 maps [p]K0 (C(X)) 7→ [p′ ]K 0 (X) .
A.2.4
Topological K-theory and cohomology
Of fundamental importance in the study of K-theory is the relationship
between the K-theory of a space and its cohomology via a homomorphism
ch : K 0 (X) → H even (X; Q) called the Chern character. The definition of the Chern
character ch(p) of a vector bundle p over X is rather involved, but it is constructed
using the Chern classes chn (p) ∈ H 2n (X; Q) and is determined by several important
functorial properties:
1. ch0 (p) = 1 for all bundles p.
87
2. If f : Y → X is a continuous map and p is a bundle over X, then ch(f ∗ p) =
f ∗ ch(p).
3. ch(p ⊕ q) = ch(p) ch(q) (cup product).
In [56], J. Villadsen used the Chern character in the noncommutative setting to lift
cohomological obstructions to obstructions in K-theory. We will employ this revolutionary technique in Chapter 6 to give a lower bound on the radius of comparison of
certain C ∗ -algebras.
A.2.5
The K1 -group
Definition A.15. Let A be a unital C ∗ -algebra and denote by U∞ (A) the direct
limit of topological

groups U(A) → U2 (A) → U3 (A) → · · · → U∞ (A) with connecting
u 0
. Define an equivalence relation ∼h on U∞ (A) by u0 ∼h u1 if u0
maps u 7→ 
0 1
is homotopic to u1 (that is, there is a continuous path ut : [0, 1] → U∞ (A) connecting
u0 and u1 ).
We define K1 (A) = U∞ (A)/ ∼h , which is an abelian group with addition [u]+[v] =
[u ⊕ v].
Observation A.1. Let u, v ∈ U∞ (A). The Whitehead Lemma states that u ⊕ v ∼h
uv ⊕ 1. Hence inverses in K1 (A) are well-defined by −[u] = [u∗ ].
Definition A.16. The suspension of a C ∗ -algebra A is defined by SA =
{f ∈ C([0, 1], A) | f (0) = f (1) = 0}. Likewise, the suspension of a topological space
X is defined as the quotient space SX = X × [0, 1]/ ∼ where (x1 , 0) ∼ (x2 , 0) and
(x1 , 1) ∼ (x2 , 1) for all x1 , x2 ∈ X.
The Bott Periodicity Theorem defines an isomorphism K0 (A) → K1 (SA) for every
C ∗ -algebra A. Defining the K 1 -group of a topological space requires more advanced
homological tools, but Bott periodicity gives an isomorphism K 1 (X) ∼
= K 0 (SX).
88
A.3
The Classification Program
The classification program for nuclear C ∗ -algebras began c. 1990 when G. Elliott
conjectured that simple separable nuclear C ∗ -algebras would be classified up to isomorphism by their K-theory and traces. In this section, we give a brief review of the
definitions and several theorems relating to the classification of C ∗ -algebras.
Definition A.17. Let A and B be C ∗ -algebras. A linear map ϕ : A → B is positive if
ϕ(A+ ) ⊂ B+ and completely positive if its inflation ϕ : Mn (A) → Mn (B) is positive for
all n ∈ N. A linear map ϕ : A → B is a contraction (or is contractive) if kϕ(a)k ≤ kak
for all a ∈ A. If ϕ : A → B is both completely positive and contractive we say it is
c.p.c.
Example A.12. Clearly, all C ∗ -homomorphisms are c.p.c. More generally, Stinespring’s Theorem shows that c.p.c. maps ϕ : A → B are of the form ϕ(a) = vπ(a)v ∗
where π is a homomorphism and v is a contraction in some C ∗ -algebra containing B.
Definition A.18. A C ∗ -algebra A is nuclear if there exists a sequence of c.p.c. maps
ψn
ϕn
A → Mk(n) → A that converge pointwise to the identity map (i.e., kϕn ◦ψn (a)−ak → 0
for all a ∈ A).
Conjecture 5 (Elliott’s Conjecture, [4, 5.1]). There is a K-theoretic functor F defined on the category of separable nuclear C ∗ -algebras such that if there is an isomorphism ϕ : F (A) → F (B), then there is a C ∗ -isomorphism Φ : A → B such that
F (Φ) = ϕ.
Although it has since been shown that the conjecture does not hold in full generality, considerable success has been made towards its resolution:
Definition A.19. A C ∗ -algebra A is approximately finite-dimensional or AF if A ∼
=
limn An where each An is a finite-dimensional C ∗ -algebra.
−→
Theorem A.8 (Elliott’s Theorem). Suppose A and B are AF algebras.
If
ϕ : K0 (A) → K0 (B) is an isomorphism of partially ordered abelian groups such that
ϕ([1A ]) = [1B ] then there exists an ∗-isomorphism π : A → B such that K0 (π) = ϕ.
89
Elliott’s Theorem led to the conjecture that all simple separable nuclear C ∗ algebras may be classified up to isomorphism by K-theoretic information. However, in [52] A. Toms constructed a family of simple, separable, nuclear C ∗ -algebras
which are non-isomorphic yet have identical K-theory and traces (in fact, are nondistinguishable with respect to any homotopy-invariant continuous functor) providing
a fundamental counterexample to Conjecture 5.
Nevertheless, Conjecture 5 has been verified for large classes of C ∗ -algebras satisfying different regularity assumptions. Several landmark examples are:
• In [26], E. Kirchberg and N.C. Phillips showed that Conjecture 5 holds for unital
purely infinite algebras satisfying the UCT.
• In [29], H. Lin verified Conjecture 5 for algebras that are tracially AF.
• In [14], G. Elliott, G. Gong, and L. Li proved that the class of unital AH algebras
with very slow dimension growth satisfy Conjecture 5.
Perhaps the most important regularity condition that has emerged recently is
Z-stability:
Theorem A.9. [25, Theorem 2.9] There exists an infinite-dimensional unital simple
C ∗ -algebra Z that has a unique tracial state, K0 (Z) ∼
= K0 (C) as scaled ordered groups,
and K1 (Z) = K1 (C) = 0. We call Z the Jiang-Su algebra.
Definition A.20. A C ∗ -algebra A is Z-stable if it absorbs the Jiang-Su algebra
tensorially: A ⊗ Z ∼
= A.
As shown in Theorem A.9, Z has the same K-theory and traces as the complex
numbers, so K∗ (A) ∼
= K∗ (A ⊗ Z) and T (A ⊗ Z) ∼
= T (A) for any C ∗ -algebra A. Hence
Z-stability should be required of any C ∗ -algebras we hope to classify up to isomorphism by K-theoretic invariants. W. Winter’s revolutionary two-step approach to the
classification program was to show that Z-stable algebras are classifiable (satisfied
Conjecture 5) and then characterize the class of C ∗ -algebras that are Z-stable. A
huge amount of progress towards the resolution of Conjecture 5 has been made using
90
this approach. In [59], Winter shows (using results of H. Lin) that a large class of
C ∗ -algebras are classified up to Z-stability, and a steadily larger class of C ∗ -algebras
has been shown to be Z-stable. Of special importance to this dissertation is Toms’
and Winter’s result about crossed products.
Theorem A.10. [55, Theorem 0.2] Let (X, α) be a minimal finite-dimensional topological dynamical system. Then the crossed product C(X) ⋊α Z is Z-stable.
Using this theorem, Toms and Winter were able to show that crossed products
coming from minimal finite-dimensional uniquely ergodic dynamical systems satisfy
Conjecture 5. In this dissertation, we hope to extend Theorem A.10 into the infinitedimensional case. We prove that a large class of crossed products (coming from
infinite-dimensional dynamical systems) have strict comparison. Theorem 1.1 of [35]
implies that these C ∗ -algebras are Z-stable in the uniquely ergodic case, bringing us
one step closer to their classification.
91
APPENDIX B. Topological Dynamical Systems
Definition B.1. By a (topological) dynamical system we mean a pair (X, α) where
X is a compact metrizable space and α : X → X is a homeomorphism. Equivalently,
a topological dynamical system is a continuous action of the integers on a compact
metrizable space X with the action defined by n · x = αn (x).
Example B.1.
1. Let X = T (viewed as a subset of C) and let θ ∈ [0, 1). Define the homeomorphism ρθ : X → X by ρθ (z) = e2πiθ z. The map ρθ can be visualized as ‘rotating’
the points on T by 2πθ radians and is often called a rotation map on T.
Q
2. Let K be any compact metrizable space and set X = n∈Z K = K Z . Define the
homeomorphism σ : X → X by σ(kn ) = (kn+1 ). Since σ is defined by shifting
indices, it is often called the shift map on X. The system (X, σ) is called the
full shift or Bernoulli shift on K.
3. More generally, any continuous action G y X of a locally compact Hausdorff
topological group G on a compact metrizable space X defines a dynamical system. However, in this dissertation we focus on the case where G = Z.
Definition B.2. A morphism f : (X, α) → (Y, β) of topological dynamical systems
is a continuous function f : X → Y that is equivariant in the sense that f ◦ α = β ◦ f .
The systems (X, α) and (Y, β) are isomorphic if f is a homeomorphism.
Definition B.3. A subset Y ⊂ X of a topological dynamical system (X, α) is invariant if α(Y ) ⊂ Y . If Y is a compact and α(Y ) = Y , the system (Y, α |Y ) is called
a subsystem of (X, α). A topological dynamical system (X, α) is called minimal if it
has no closed invariant subsets other than ∅ and X itself.
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Remark B.1. There are many equivalent definitions of minimality. For example, it
is easy to see that (X, α) is minimal if and only if the orbit orbit(x) = {αn (x)}n∈Z is
dense in X for any x ∈ X.
Example B.2. If θ ∈ [0, 1) is irrational, then the rotation system (T, ρθ ) is minimal. To see this, identify T with R/Z and ρθ : T → T with the translation map
τθ : R/Z → R/Z defined by x 7→ x + θ (mod Z). If θ is irrational, then τθ has infinite
order and the orbit of every point is dense in R/Z.
As noted by G.D. Birkhoff, the collection of all subsystems of a dynamical system
(X, α) is partially ordered by inclusion and a straightforward application of Zorn’s
Lemma implies that every dynamical system contains a minimal subsystem. However,
it is not always the case that every topological dynamical system can be decomposed
as the disjoint union of its minimal subsystems (this is only true when the restriction
of α to each orbit closure is minimal). Nevertheless, it is possible to construct useful
decompositions of topological dynamical systems as shown in the following section.
B.1
Decomposition of Dynamical Systems
The Rokhlin Lemma is a fundamental result in measurable dynamics that gives a
decomposition of an arbitrary (aperiodic) measurable dynamical system into cycles.
Theorem B.1 (Rokhlin Theorem). Let (X, α) be a topological dynamical system
and let µ be an α-invariant probability measure on X. Assume that the µ-measure
of the set of periodic points is zero. Then for any ǫ > 0 and any n ∈ N there
exists a subset B ⊂ X such that B, α(B), . . . , αn−1 (B) are pairwise disjoint and
µ(∪ni=0 αi (B)) > 1 − ǫ.
For our purposes, we need a continuous (as opposed to measurable) analogue of
this decomposition theorem.
Definition B.4. If (X, α) is a minimal system and K ⊂ X is a closed set with
nonempty interior, then the orbit of every point x ∈ X intersects K. In particular,
93
for x ∈ K we call the integer nK (x) = inf{n > 0 | αn (x) ∈ K} the first return time
of x to K.
Since nK : K → N is lower semicontinuous, a straightforward Baire Category argument shows that the number of first return times is finite, say im(nK ) = {n1 , . . . , nℓ }.
Therefore we can partition K into subsets K1 , . . . , Kℓ where Ki = {x ∈ K | nK (x) =
ni }. By lower semicontinuity, K1 is closed, as is K1 ∪ · · · ∪ Ki for any i ≤ ℓ. However,
Ki is not closed in general.
Definition B.5. Let (X, α) be a minimal system and K ⊂ X a closed set with
nonempty interior. By Definition B.4, we can partition K = K1 ⊔ K2 ⊔ · · · ⊔ Kℓ by
first return time. In turn, this induces a decomposition of the entire space: X =
i −1
⊔ℓi=1 ⊔nj=0
αj (Ki ). We call this the Rokhlin tower decomposition of X induced by K,
which consists of ℓ Rokhlin towers, where the i-th tower has height ni .
n−1 i
Definition B.6. If (X, α) can be decomposed into a single cycle X = ⊔i=0
α (K)
where K ⊂ X is compact and αn (K) = K we call this a perfect Rokhlin tower.
Example B.3. As shown in Lemma B.2, odometers have many perfect Rokhlin tower
decompositions.
B.2
Cantor Minimal Systems
A Cantor minimal system is a minimal dynamical system on a Cantor space.
Odometers are special examples of Cantor minimal systems with especially nice
Rokhlin tower decompositions. All odometers have mean dimension zero (in fact,
since the underlying topological space is a Cantor set, have covering dimension zero)
but their cyclic decompositions make them an extremely useful.
Definition B.7. Let (mk ) be a sequence of integers with each mk ≥ 2, and define
the finite discrete spaces Yk = {0, . . . , mk − 1}. This sequence defines a Cantor space
Q
Y = k Yk (with the product topology). The odometer action β : Y → Y is defined
by β(. . . , y3 , y2 , y1 ) = (. . . , y3 , y2 , y1 ) + (. . . , 0, 0, 1) with ‘carryover’ addition, so that
94
(. . . , 0, 0, m1 − 1) + (. . . , 0, 0, 1) = (. . . , 0, 1, 0) and (. . . , m3 − 1, m2 − 1, m1 − 1) +
(. . . , 0, 0, 1) = (. . . , 0, 0, 0) (hence the name, since the action mimics the cycling of an
automobile’s odometer).
Alternatively, the dynamical system (Y, β) can be defined as the inverse group
limit Y = lim Z/m1 · · · mk (under the profinite topology) with β(y) = y + 1.
←−
Example B.4. The most commonly studied example of an odometer system is the
Q
2-adic (or 2∞ ) odometer, defined on the space ∞
k=1 {0, 1}. Equivalently, this system
can be viewed as the 2-adic integers Z2 = lim Z/2k with the action given by x 7→ x+1.
←−
Lemma B.2. Let (Y, β) be an odometer. Then there exists a sequence En of partitions
of Y with the following properties:
n −1
1. Each family En = {Ei,n }Ii=0
is a finite partition of Y into compact-open subsets.
2. The map β acts cyclically on each En , that is, β(Ei,n ) = Ei+1
(mod In ),n .
3. Every set Ei,n ∈ En is a disjoint union of elements of En+1 .
4. With respect to any metric on Y , diam(Ei,n ) → 0 as n → ∞.
Proof. Using the first part of Definition B.7, we have that Y =
Q
k
Yk where each
Yk = {0, . . . , mk − 1}. The partitions En can now be defined inductively. Set E0 = Y ,
Q
Q
Q
E1 = { k>1 Yk × {0}, k>1 Yk × {1}, . . . , k>1 Yk × {m1 − 1}}, and continue in this
Q
Q
fashion so that in general En = { k>n Yk × {0} × {0} × · · · × {0}, k>n Yk × {0} ×
Q
{0}×· · ·×{1}, . . . , k>n Yk ×{mn −1}×{mn−1 −1}×· · ·×{m1 −1}}. The conditions
follow easily from this definition.
Remark B.2. The second condition of Lemma B.2 implies that (Y, β) can be decomposed into a perfect Rokhlin tower of height In . Thus Corollary 3.12 shows that
C(Y ) ⋊β Z has a nice decomposition as well, specifically, C(Y ) ⋊β Z ∼
= MIn (E0,n ).
We make extensive use of this fact in Chapter 6.
95
B.3
Disjoint Systems
In Theorem 1.2, we consider dynamical systems whose product is minimal, begging
the question of which systems fulfill this property. This question is classical (see [17])
and is centered in the study of disjoint dynamical systems.
Definition B.8. A joining of two topological dynamical systems (X, α) and (Y, β) is
a closed invariant subset of the product system (X × Y, α × β) that projects onto both
X and Y . The systems (X, α) and (Y, β) are disjoint if their only joining is X × Y .
Example B.5. No dynamical system is disjoint from itself, since the diagonal subset
is closed, invariant, and projects onto both factors.
Observation B.1. If (X, α) and (Y, β) are minimal and disjoint then their product
(X × Y, α × β) is minimal by definition. Although this provides many examples of
minimal product systems, here we only focus on the case of distal and weakly mixing
systems.
Definition B.9. A dynamical system (X, α) is distal if for any choice of metric d on
X and any x0 , x1 ∈ X, there exists ǫ > 0 such that d(αn (x0 ), αn (x1 )) ≥ ǫ for all n.
Example B.6.
1. The rotation maps of Example B.1.1 are distal since they are isometric with
respect to the usual absolute value.
2. Every odometer is distal. In fact, a Cantor minimal system is distal if and only
if it is an odometer.
Definition B.10. A dynamical system (X, α) is (topologically) transitive if for any
two nonempty open sets U, V ⊂ X, there exists n ∈ N such that αn (U ) ∩ V 6= ∅. The
system is weakly mixing if the product (X × X, α × α) is transitive.
Lemma B.3. [17, Theorem II.3] Distal systems are disjoint from weakly mixing systems. Therefore if (X, α) is a minimal weakly mixing system and (Y, β) is a minimal
distal system, the product system (X × Y, α × β) is minimal.
96
Weakly mixing systems abound. All Bernoulli shifts are weakly mixing; in fact,
any minimal system is either weakly mixing or else is particularly rigid (has a nontrivial isometric factor) (see e.g. [49, Theorem 2.7.12]). Therefore, the product of an
arbitrary minimal system with a Cantor odometer is very often minimal (a fact which
will be important in Theorem 1.2).
97
APPENDIX C. Crossed Product C ∗ -Algebras
In this section we give a brief review of crossed product C ∗ -algebras associated to
topological dynamical systems, borrowing heavily from many standard references.
For a more thorough review of crossed products see e.g. [7, Section 4.1], [12, Chapter
VIII], or [42].
Definition C.1. Let A be a C ∗ -algebra and ϕ : A → A an automorphism. A covariant representation of (A, ϕ) on a Hilbert space H is a homomorphism π : A → B(H)
together with a unitary operator u ∈ U(H) such that π(ϕ(a)) = uπ(a)u∗ for all a ∈ A.
Example C.1. Let (X, α) be a topological dynamical system. Then α defines an
automorphism ϕα on the C ∗ -algebra C(X) by ϕα (f ) = f ◦α−1 . Let µ be any invariant
measure on (X, α) (invariant, in fact ergodic, measures always exist; see e.g. [12, Theorem VIII.3.1, Proposition VIII.3.2]). This measure induces a covariant representation
of C(X) on B(L2 (X, µ)), where πµ : C(X) → B(L2 (X, µ)) is represented as multiplication operators and u ∈ U(L2 (X, µ)) is defined by u(f ) = f ◦ α−1 . It is easy to
check that u is unitary (using the fact that µ is invariant). Finally, for a ∈ C(X) and
f ∈ L2 (X, µ),
πµ (ϕα (a))(f (x)) = (ϕα (a))(x)f (x)
= a(α−1 (x))f (x)
= u(a(x)f (α(x)))
= (ua)(f (α(x)))
= (uau∗ )(f (x)).
Definition C.2. The crossed product C ∗ -algebra corresponding to the topological dynamical system is the universal C ∗ -algebra for covariant representations of (C(X), ϕα )
98
and is denoted C(X)⋊α Z (the notation comes from the fact that ϕα defines a Z-action
on C(X) by n · a = ϕnα (a)).
Theorem C.1. [12, Corollary VIII.3.6] If µ is an ergodic measure on (X, α) and
X is infinite then C(X) ⋊α Z is isomorphic to the sub-C ∗ -algebra of B(L2 (X, µ))
generated by πµ (C(X)) and u.
Example C.2.
1. Let (X, id) be the identity system on a compact space X. Then C(X) ⋊id Z
is generated by C(X) ⊂ B(L2 (X)) and a unitary commuting with C(X). By
the Spectral Theorem, the C ∗ -algebra generated by a single unitary (with full
spectrum) is isomorphic to C(S 1 ), and hence C(X) ⋊id Z ∼
= C(X) ⊗ C(S 1 ) ∼
=
C(X × S 1 ).
2. Define the n-periodic dynamical system (S 1 , ρn ) by ρn : z 7→ e2πi/n z. As shown
in [42, Example 10.9], C(S 1 ) ⋊ρn Z ∼
= Mn (C(S 1 )).
Remark C.1. Identifying C(X) with πµ (C(X)) in B(L2 (X, µ)), it is convenient to
P
n
view C(X) ⋊α Z as the closure of the set of elements of the form N
n=−N fn u , where
fn ∈ C(X). In this dissertation we make extensive use of the fact that finite sums of
this form are dense in C(X) ⋊α Z.
Lemma C.2. There exists a faithful expectation E : C(X) ⋊α Z → C(X) such that
P
E( n fn un ) = f0 .
Theorem C.3. If (X, α) is minimal and X is infinite, then any ergodic measure
R
µ induces a faithful trace on C(X) ⋊α Z given by τ (a) = X E(a) dµ. If (X, α) is
uniquely ergodic, then the trace is unique.
Observation C.1. In general, any invariant measure µ on (X, α) corresponds to a
R
trace on C(X) ⋊α Z given by τ (a) = X E(a) dµ. However, such a trace will not be
faithful in general.
Theorem C.4. The crossed product C(X) ⋊α Z is simple if and only if (X, α) is
minimal.
99
idea. Roughly speaking, a closed invariant subspace Y of (X, α) corresponds to an
ideal IY generated by {f ∈ C(X) | f |Y = 0} and conversely.
100
APPENDIX D. The Cuntz Semigroup
In this Appendix we give a brief overview of the Cuntz Semigroup of a C ∗ -algebra.
For a more in-depth introduction see [4, 10].
Definition D.1. Let A be a C ∗ -algebra, and let a, b ∈ (A ⊗ K)+ . We say that a is
Cuntz subequivalent to b (written a . b) if there exists a sequence xn ∈ A ⊗ K such
that ka − xn bx∗n k → 0. We say that a is Cuntz equivalent to b (written a ∼ b) if a . b
and b . a.
The Cuntz semigroup Cu(A) is the collection of all positive elements in A ⊗ K
modulo Cuntz equivalence. This collection is an ordered abelian semigroup, with
addition given by


a 0

[a] + [b] = 
0 b
and an order structure given by
[a] ≤ [b] if a . b.
It is straightforward to check that the addition and ordering are well-defined and
compatible with each other (in the sense that [a0 ] ≤ [a1 ] implies [a0 ] + [b] ≤ [a1 ] + [b].)
Example D.1.
1. If A = C, then Cuntz (sub)equivalence reduces to rank (that is, [a] ≤ [b] if and
only if dim aK ≤ dim bK). Therefore Cu(A) = {0, 1, . . . , ∞} with the normal
addition and order structure.
2. If p, q ∈ A ⊗ K are projections, then [p] . [q] if and only if p is Murray-von
Neumann subequivalent to q [4, Lemma 2.18]. This shows that the Murray-von
Neumann semigroup V (A) is in a sense ‘contained’ in the Cuntz semigroup (at
the very least there is a natural map V (A) → Cu(A) given by [p]M-vN 7→ [p]Cu ).
101
Although it is not the case in general, for stably finite algebras this map V (A) →
Cu(A) is an order embedding by [4, Lemma 2.20].
3. If A is a purely infinite simple C ∗ -algebra, then [28, 2.2 Theorem (b)] shows that
a . b whenever a and b are nonzero. Hence Cu(A) = {0, ∞}.
4. If B is a subalgebra of A, then there is a natural map Cu(B) → Cu(A). Given
positive elements a, b ∈ B ⊗ K ⊂ A ⊗ K, we write a .B b and a .A b for Cuntz
subequivalence in Cu(B) and Cu(A), respectively. Of course, a .B b implies
a .A b but not conversely.
Lemma D.1. [4, Lemma 2.20] If A is stably finite, then the natural map
V (A) → Cu(A) defined by [p]0 7→ [p]Cu is an order embedding.
Definition D.2. [4, Definition 2.4] Let A be a C ∗ -algebra and a ∈ (A ⊗ K)+ . For
ǫ > 0, define the function f : σ(a) → R by f (x) = max{0, x − ǫ}. By the Spectral
Theorem, this defines a positive element f (a) ∈ A ⊗ K which we denote (a − ǫ)+ .
Example D.2.
1. Using functional calculus, ((a − ǫ)+ − ǫ′ )+ = (a − ǫ − ǫ′ )+ .
2. By [4, Corollary 2.9], (a − ǫ)+ . a.
3. By [4, Theorem 4.2], if ka − bk < ǫ then (a − ǫ)+ . b.
Definition D.3. If A is a C ∗ -algebra, then any 2-quasitrace τ ∈ QT2 (A) induces a
state on the Cuntz semigroup dτ : Cu(A) → [0, ∞] defined by dτ ([a]) = limn τ (a1/n ).
Lemma D.2. If a, b ∈ (A ⊗ K)+ and a . b then dτ ([a]) ≤ dτ ([b]) for any τ .
D.1
The Cuntz Semigroup from an A-module perspective
The Cuntz semigroup is a way to generalize the study of K-theory for C ∗ -algebras
(see Appendix A). As explained in Section 3.2, in the commutative case (or when
studying topological K-theory for smooth manifolds) K-theory is built on the study
102
of vector bundles. By the Serre-Swam Theorem, vector bundles over a manifold X
are equivalent to finitely-generated projective C ∞ (X)-modules, and the K-theory of
the manifold X is built out of these objects (see Section A.2.2).
In this section, we show how the Cuntz semigroup generalizes this process to
projective A-modules. Building a semigroup out of A-modules makes it possible to
gather cohomological invariants of C ∗ -algebras even though no ‘vector bundles’ exist.
In particular, the Cuntz semigroup is perfectly useful in the absence of nontrivial
projections, which is of course not the case for the K-groups.
Definition D.4. Let A be a C ∗ -algebra. An A-module E is countably generated if
there exists a countable subset {en }∞
n=0 ⊂ E whose A-linear span is E.
We say that E is a Hilbert A-module if it has an A-valued inner product
h·, ·i : E × E → A such that for all e1 , e2 , e3 ∈ E, z, w ∈ C, and a ∈ A:
1. he1 , αe2 + βe3 i = αhe1 , e2 i + βhe1 , e3 i,
2. he1 , e2 ai = he1 , e2 ia,
3. he1 , e2 i = he2 , e1 i∗ ,
4. he1 , e1 i ≥ 0, and he1 , e1 i = 0 if and only if e1 = 0,
and furthermore E is complete with respect to the norm kek2 = khe, eik.
Example D.3.
1. A closed ideal I of A is a Hilbert A-module with inner product given by ha, bi =
a∗ b. In particular, if a ∈ A is a positive element, then the ideal aA ⊂ A is a
Hilbert A-module.
2. If E1 , . . . , En are Hilbert A-modules, then their direct sum E1 ⊕ · · · ⊕ En is a
P
Hilbert A-module with inner product h(ai ), (bi )i = i hai , bi i.
3. If E1 , E2 , . . . is a family of Hilbert A-modules, their direct sum ⊕n En = {(en ) ∈
Q
P
n En |
n hen , en i converges in A.} is a Hilbert A-module with inner product
P
h(en ), (e′n )i = n hen , e′n i. A special case is the Hilbert A-module HA = ⊕∞
n=1 A
(which is also countably generated).
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Theorem D.3. [4, Proposition 3.10] Every finitely generated projective A-module is
isomorphic to a Hilbert A-module.
Definition D.5. Let A be a C ∗ -algebra and E and F Hilbert A-modules. Denote by
BA (E, F ) the set of adjointable operators T : E → F , such that there exists a unique
operator T ∗ : F → E satisfying hT e, f i = he, T ∗ f i for all e ∈ E, f ∈ F .
Example D.4. If e ∈ E and f ∈ F , we can define an analogue of a rank-one linear
map over A by θf,e (x) = f he, xi. Notice that these operators satisfy the properties:
∗
1. θf,e
= θe,f ,
2. θf,e θe′ ,f ′ = θehf,f ′ i,e′ = θe,e′ hf ′ ,f i ,
3. T θf,e = θT f,e for T ∈ BA (F, G), and θf,e S = θf,S ∗ e for T ∈ BA (G, E).
Definition D.6. Define the subalgebra of compact operators KA (E, F ) ⊂ BA (E, F )
as the closed linear span of the set {θf,e | f ∈ F, e ∈ E}. Denote by KA (E) =
KA (E, E).
Example D.5.
1. KA (A) = A.
2. KA (An ) = Mn (A).
3. KA (HA ) = A ⊗ K (where K here is the usual compact operators on Hilbert
space).
Definition D.7. Let E be a Hilbert submodule of the Hilbert A-module F . We say
that E is compactly contained in F , written E ⊂⊂ F , if there exists a self-adjoint
operator T ∈ KA (F ) such that T |E = 1 |E .
Definition D.8. Let E and F be Hilbert A-modules. We say that E is Cuntz
subequivalent to F , written E . F , if every compactly contained submodule E0 ⊂⊂ E
is isometrically isomorphic to some F0 ⊂⊂ F . If both E . F and F . E we say that
E and F are Cuntz equivalent (written E ∼ F ).
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Example D.6. By [4, Lemma 4.8], if E ⊂ F then E . F .
Definition D.9. Denote by Cu(A) the set of isomorphism classes of countablygenerated Hilbert A-modules modulo Cuntz equivalence. This set is a partially ordered abelian semigroup with addition [E] + [F ] = [E ⊕ F ].
Theorem D.4. [4, Theorem 4.33], [10, 6. Appendix] The definitions of Cu(A)
given in Definition D.1 and Definition D.9 are isomorphic via the assignment [a] 7→
[a(A ⊗ K)].
VITA
105
VITA
Taylor Hines was born in Bellevue, Washington, in 1986. He obtained an undergraduate degree in mathematics at Arizona State University in Tempe, Arizona
before coming to Purdue University for his Ph.D. in Mathematics. He graduated in
August of 2015.