Proceedings of the International Congress of Mathematicians
Hyderabad, India, 2010
Group Actions on Operator Algebras
Masaki Izumi∗
Abstract
We give a brief account of group actions on operator algebras mainly focusing on
classification results. We first recall rather classical results on the classification
of discrete amenable group actions on the injective factors, which may serve as
potential goals in the case of C ∗ -algebras for the future. We also mention Galois
correspondence type results and quantum group actions for von Neumann algebras. Then we report on the recent developments of the classification of group
actions on C ∗ -algebras in terms of K-theoretical invariants. We give conjectures
on the classification of a class of countable amenable group actions on Kirchberg
algebras and strongly self-absorbing C ∗ -algebras, which involve the classifying
spaces of the groups.
Mathematics Subject Classification (2010). Primary 46L40; Secondary 46L35.
Keywords. Operator algebras, group actions, K-theory
1. Introduction
There are two classes of main objects in the theory of operator algebras, C*algebras and von Neumann algebras. They are subalgebras of the set of bounded
operators B(H) on a complex Hilbert space H closed under the adjoint operation, and closed under appropriate topologies, the norm topology for the former,
and the weak operator topology for the latter. Since the celebrated GelfandNaimark theorem says that any abelian C ∗ -algebra with unit is isomorphic to
the set of continuous functions C(X) on a compact Hausdorff space X, C ∗ algebras are sometimes regarded as noncommutative analogues of topological
spaces, while von Neumann algebras are regarded as those of measure spaces
for a similar reason. To certain extent, this analogy is helpful to understand
the difference between the two classes, though it may be misleading sometimes.
If one further pursues the analogy, the difference between group actions on
∗ Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan.
E-mail: [email protected]
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C ∗ -algebras and those on von Neumann algebras could be compared to that
between topological dynamics and ergodic theory. Group actions on operator
algebras have always been one of the main interests in the field for the sake of
applications to physics, and of course, for intrinsic reasons.
In operator algebraic formulation of quantum physics, the symmetries
and time evolution of a physical system are usually described by (anti)automorphisms of a relevant operator algebra. Indeed, for the purpose of direct
applications to physics, a group of mathematical physicists started working on
group actions on operator algebras in the 60s, which brought important new
ideas to the field such as the KMS-condition for actions of the real numbers R
(see [1],[3],[11] for example).
Group actions are also essential for understanding the structure of operator
algebras. One can see it typically in Connes’s classification of injective factors,
which involves the classification of cyclic group actions. Factors are von Neumann algebras with trivial center, and they are building blocks of general von
Neumann algebras. Connes completely classified injective factors up to isomorphism in the mid 70s except for one case, which was settled by Haagerup about
10 years later. Thanks to these results, it turns out that injectivity, which is a
functional analytic property, is equivalent to approximately finite dimensionality (see [52]).
Connes’s argument for the classification of cyclic group actions is based on
a noncommutative analogue of the Rohlin tower construction in ergodic theory,
which has a great influence on other classification results of group actions.
A far reaching generalization of Connes’s classification of cyclic group actions
was accomplished by many hands, which says that countable amenable group
actions on injective factors are completely classified up to cocycle conjugacy by
computable classification invariants (see [23]).
These results on injective factors and countable amenable group actions on
them are one of the most significant establishments in the theory of operator
algebras, which may suggest possible goals in other areas of operator algebras
for the future. One of the purposes of this note is to report on the progress
of group actions on operator algebras after these results. We mainly focus on
the case of C ∗ -algebras, though we also mention other topics such as Galois
correspondence for compact group actions and quantum group actions in the
von Neumann algebra case.
Nuclearity for C ∗ -algebras is the right counterpart of injectivity for von
Neumann algebras. The classification of simple nuclear C ∗ -algebras is still an
ongoing project, which is called the Elliott program. Being noncommutative
topological spaces, C ∗ -algebras are expected to have classification invariants
with a topological flavor. Indeed, Elliott’s conjecture says that separable simple
nuclear C ∗ -algebras in certain classes should be classified up to isomorphism
by invariants coming from K-theory. Those classes for which the conjecture
is verified are said to be classifiable. Thanks to the remarkable progress of the
Elliott program in these two decades, there are a few known classes of classifiable
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Masaki Izumi
C ∗ -algebras with good intrinsic characterizations. Kirchberg algebras form one
of such classes with the best permanence properties; for example, they are
closed under taking crossed products by outer actions of countable amenable
groups (see [28],[47]).
Now we ask the following question: what are plausible statements for the
classification of group actions on classifiable C ∗ -algebras? Since amenability for
groups is the same sort of property as injectivity for von Neumann algebras and
nuclearity for C ∗ -algebras, we should keep the amenability assumption for the
groups in the case of classifiable C ∗ -algebras too. Although finite groups clearly
form the tamest subclass of amenable groups as far as analysis is concerned,
it is known that K-theory for finite group actions are practically out of control. Since K-theory is the most essential element in the Elliott program, finite
groups are not in a preferable situation. Indeed, recent developments of the classification of Z2 -actions on classifiable C ∗ -algebras suggest that the topology of
the automorphism groups of the C ∗ -algebras and the classifying spaces of the
groups should be involved in the classification invariants of the actions, which is
a completely new aspect from the case of injective factors. This would be rather
natural for “noncommutative topological spaces”, and would be consistent with
the difficulties in finite group actions because their classifying spaces are always
infinite dimensional. Trying to answer the above question, we formulate a few
conjectures at the end of this note.
Throughout the note, separability or second countability for topological
spaces is often assumed without mentioning it. Our standard references are
[50], [51], [52] for von Neumann algebras, [2] for K-theory, and [28], [47] for
the classification of nuclear C ∗ -algebras. The reader is referred to them for the
definitions of undefined terms, and the proofs of results stated without citing
references.
We end this section with recalling the basic definitions for group actions
on operator algebras. Let A be a C ∗ -algebra or a von Neumann algebra, and
let G be a locally compact group. An action α of G on A is a continuous
homomorphism from G to the automorphism group Aut(A). We denote by
A oα G the crossed product of A by α, which is an analogue of a semidirect
product in group theory (since we deal with only amenable groups, we do not
need to distinguish the reduced crossed products from the full crossed products).
We denote by Aα the fixed point subalgebra of A for α. Two actions α and β
of G on A are conjugate if there exists γ ∈ Aut(A) satisfying γ ◦ αg ◦ γ −1 = βg
for all g ∈ G. We denote by U (A) the group of all unitaries in A. An αcocycle u is a continuous map from G to U (A) satisfying the 1-cocycle relation
ugh = ug αg (uh ) for all g, h ∈ G. When u is an α-cocycle, one can perturb α
by u, and αgu = Ad ug ◦ αg is again an action, where Ad v denotes the inner
automorphism of A induced by v ∈ U (A). This perturbation is often allowed for
the purpose of applications because there exists an isomorphism between the
crossed products by α and αu extending the identify of A. Two actions α and
β are cocycle conjugate if there exists an α-cocycle u such that β is conjugate
Group Actions on Operator Algebras
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to αu . A G-action α is said to be outer if αg is not inner for any g ∈ G \ {e}.
In the case of C ∗ -algebras, we denote by αs the stabilization of α, which is the
G-action on A ⊗ K defined by αgs = αg ⊗ Ad π(g), where K is the set of compact
operators on `2 and π is a direct sum of infinitely many copies of the regular
representation of G.
2. Group Actions on Factors
2.1. Injective factors. We first recall the basics of injective factors. Factors are divided into type I, type II1 , type II∞ , and type III. A type I factor
is isomorphic to B(H) for a Hilbert space H, and so it is completely characterized by the dimension of H. A type II1 factor M is characterized as an
infinite dimensional factor with a finite trace τ , a linear functional satisfying
τ (ab) = τ (ba) for all a, b ∈ M and τ (1) = 1. A type II∞ factor is isomorphic to the tensor product of a type II1 factor and B(`2 ), and it has a unique
(unbounded) semifinite trace up to a scalar multiple. Remaining factors are of
type III.
We give fundamental examples of nuclear C ∗ -algebras and injective factors
here. Let {nj }∞
j=1 be a sequence of natural numbers greater than 1. For k ∈ N,
we set
k
O
Mnj (C),
Ak =
j=1
where Mn (C) denotes the matrix algebra over C. Since Mn (C) is identified
with B(Cn ), it is a C ∗ -algebra. We embed Ak into Ak+1 by ιk : Ak 3 x 7→
x ⊗ 1 ∈ Ak+1 . Since this embedding is isometric, the inductive limit of the
system {Ak }k∈N has a norm extending that of Ak . The completion A of the
inductive limit with respect to this norm is called the UHF-algebra, a typical
example of a simple nuclear C ∗ -algebra. Let τk be the normalized trace of
Ak , which is the usual trace divided by n1 n2 · · · nk . Since the restriction of
τk+1 to Ak coincide with τk , there exists a trace τ of A extending τk . We can
introduce an inner product into A by hx, yi = τ (y ∗ x), and we denote by Hτ
the completion of A with respect to this inner product. The UHF-algebra A
acts on Hτ by left multiplication (called the GNS representation for τ ), and the
weak closure R of A in this representation is an injective type II1 factor. While
Murray and von Neumann showed that the isomorphism class of R does not
really depend on the sequence {nj }∞
j=1 , Glimm [10] completely classified the
UHF-algebras up to isomorphism whose classification invariant is the formal
product of {nj }∞
j=1 , called the supernatural number. This means that there
are infinitely many isomorphism classes of UHF-algebras, and one can see the
sharp contrast between the two theories here. If we replace τ with other product
states, we can obtain factors of other types, called Araki-Woods factors.
Type III factors are further divided into type IIIλ , 0 ≤ λ ≤ 1, thanks to
Tomita-Takesaki theory. Every type IIIλ factor with 0 < λ < 1 is expressed
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Masaki Izumi
as the crossed product N oα Z with a type II∞ factor N and a trace-scaling
Z-action α. Likewise, every type III1 factor is expressed as N oα R.
The classification of injective factors says that there exists a unique isomorphism class of injective factors for each type except for the type III0 case. The
injective type III0 factors are in one-to-one correspondence with the nontransitive ergodic flows.
2.2. The classification of group actions. Although type I factors
are considered as rather trivial objects in operator algebras, we nevertheless
start with group actions on them in order to illustrate the difference between
the two equivalence relations, conjugacy and cocycle conjugacy. It is known that
every automorphism of a type I factor is inner, which shows that an action of a
countable group Γ on a type I factor B(H) is a synonym of a projective unitary
representation of Γ in H. Therefore the classification of Γ-actions on B(H) up
to conjugacy is equivalent to that of projective unitary representations of Γ in
H up to unitary equivalence, which cannot be accomplished even for Γ = Z2
and H = `2 . On the other hand, the set of cocycle conjugacy classes of Γ-actions
on B(`2 ) is in one-to-one correspondence with the second cohomology group
H 2 (Γ, T), which is a computable object.
To a Γ-action α on a type II∞ factor, one can associate a homomorphism
mα : Γ → R×
+ by the relation τ ◦ αg = mα (g)τ , where τ is the unique (up to a
scalar multiple) semifinite trace τ on the II∞ factor.
Connes [5], Jones [21], and Ocneanu [42] obtained a complete classification
result of countable amenable group actions on the injective type II factors. For
simplicity, we state the following particular case here.
Theorem 2.1. Let Γ be a countable amenable group.
(1) There exists a unique cocycle conjugacy class of outer Γ-actions on the
injective type II1 factor.
(2) For a given homomorphism m : Γ → R×
+ , there exists a unique cocycle conjugacy class of outer Γ-actions α on the injective type II∞ factor
satisfying mα = m.
A countable group Γ is amenable if there exists a left-invariant linear functional ϕ on `∞ (Γ) with ϕ(1) = ||ϕ|| = 1. For example, every solvable group is
amenable. The amenability assumption in the above theorem is known to be
necessary.
General (not necessarily outer) Γ-actions α on the injective II1 factor are
classified up to cocycle conjugacy by a relative cohomology type invariant
with respect to the normal subgroup N (α) = {g ∈ Γ| αg is inner}. Countable amenable group actions on injective type III factors are also completely
classified up to cocycle conjugacy due to Katayama, Kawahigashi, Sutherland,
and Takesaki (see [23] for the final form of the statement).
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While the original proofs of these results are a massive collection of case-bycase analysis depending on types, Masuda [33] recently obtained a short proof,
which is independent of types. Interestingly enough, his proof is based on Evans
and Kishimoto’s intertwining argument ([9]) developed in C ∗ -algebras.
An action α of a compact group G on a factor M is said to be minimal if α
is faithful as a homomorphism from G to Aut(M ), and the relative commutant
of the fixed point subalgebra M ∩ M α 0 = {x ∈ M | ∀y ∈ M α , xy = yx} is
trivial. The following theorem is obtained by Popa and Wassermann [46] for
compact Lie groups, and by Masuda and Tomatsu [34] for general compact
groups.
Theorem 2.2. For every separable compact group G, there exists a unique
conjugacy class of minimal G-actions on the injective type II1 factor and type
II∞ factor.
As a discrete Kac algebra, the dual of a compact group is amenable. Masuda
and Tomatsu actually generalized Theorem 2.1 to actions of Kac algebras, and
they obtained Theorem 2.2 by a duality argument.
The full classification of minimal actions of compact groups on the injective
type III factors is still in progress. See [14], [35] for related results.
2.3. Galois correspondence. The first Galois correspondence type result for group actions on operator algebras was obtained by Nakamura and
Takeda [41] in 1960 for finite group actions on II1 -factors. The following generalization was obtained by Longo, Popa and the author [17].
Theorem 2.3. Let α be a minimal action of a compact group G on a factor
M . Then there exists a one-to-one correspondence between the closed subgroups
H ⊂ G and the intermediate subfactors M α ⊂ N ⊂ M given by H 7→ M α|H =
N.
To obtain this result, more general inclusions of factors are studied in [17],
which has applications to algebraic quantum field theory (see [22] for example).
The crossed product inclusion of an outer action of a countable group on a factor
is also a particular case.
Theorem 2.4. Let α be an outer action of a countable group Γ on a factor M .
Then there exists a one-to-one correspondence between the subgroups Λ ⊂ Γ
and the intermediate subfactors M ⊂ N ⊂ M oα Γ given by Λ 7→ M oα
Λ = N.
Tomatsu [54] generalized Theorem 2.3 to arbitrary compact quantum groups
overcoming technical difficulties due to the lack of normal conditional expectations onto intermediate subfactors. Left coideals play the role of closed subgroups in this case.
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Masaki Izumi
2.4. Quantum group actions and Poisson boundaries. Let
G be a closed subgroup of the unitary group U (n), which acts on the matrix
algebra Mn (C) by conjugation. The infinite tensor product (ITP) of this action
is a typical example of a minimal action of G on an injective factor. A similar
construction works for a compact quantum group too if the right order of the
tensor products is chosen. However, if the antipode of the quantum group is
not involutive, (for example, it is the case for the q-deformations of the classical
groups), the ITP action is never minimal. What is the relative commutant of the
fixed point algebra then? It turns out that the relative commutant is identified
with the noncommutative Poisson boundary for a convolution operator acting
on the dual quantum group.
The notion of noncommutative Poisson boundaries was introduced by the
author in [13] in order to answer the above question. Let M be a von Neumann algebra, and let P : M → M be a unital normal complete positive map.
Although the fixed point set H ∞ (M, P ) = {x ∈ M | P (x) = x} is not necessarily an algebra, one can introduce a new product into H ∞ (M, P ) so that it
becomes a von Neumann algebra. We call the von Neumann algebra obtained
in this way the noncommutative Poisson boundary for the pair (M, P ). When
M is commutative, the new product of H ∞ (M, P ) is commutative too, and
H ∞ (M, P ) is identified with the L∞ -space over the Poisson boundary of the
random walk given by the Markov operator P .
In the case of the q-deformations of SU (N ), Neshveyev, Tuset and the
author [19] showed that the noncommutative Poisson boundary is identified
with the quantum flag manifold SUq (N )/TN −1 , which is generalized to the qdeformations of arbitrary classical groups by Tomatsu [53]. For related results
on other quantum groups, see Vaes and Vander Vennet [55], [56].
3. Group Actions on C ∗ -algebras
3.1. K-theory. We first recall the basics of K-theory, which gives efficient
isomorphism invariants of C ∗ -algebras. K-theory of C ∗ -algebras is a functor
from the category of C ∗ -algebras to that of abelian groups, which associates
two abelian groups K0 (A) and K1 (A) to a C ∗ -algebra A.
We denote by Mn (A) the C ∗ -algebra of the n by n matrices with entries in
A, by Pn (A) the set of projections in Mn (A), and by Un (A) the set of unitaries
in Mn (A). For x ∈ Mm (A) and y ∈ Mn (A), we set
x 0
x⊕y =
∈ Mm+n (A).
0 y
We say that two projections p, q in a C ∗ -algebra A are equivalent if there
exists v ∈ A such that v ∗ v = p and vv ∗ = q. We say that p and q in Pm (A)
are stably equivalent if there exists r ∈ Pn (A) such that p ⊕ r and q ⊕ r are
equivalent in Mm+n (A). Identifying p ∈ Pm (A) with p ⊕ 0 ∈ Pm+1 (A), we
Group Actions on Operator Algebras
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S∞
regard Pm (A) as a subset of Pm+1 (A), and we set P∞ (A) = m=1 Pm (A).
We denote by K0 (A)+ the set of stable equivalence classes of the projections
in P∞ (A), which is a semigroup with addition given by the direct sum. Its
Grothendieck group is K0 (A). The element in K0 (A) given by the equivalence
class of p ∈ Pm (A) is denoted by [p]0 .
Identifying a unitary u ∈ Um (A) with u ⊕ 1 ∈ Um+1 (A), we regard
Um (A) as a closed subgroup of Um+1 (A), which induces a homomorphism from
Um (A)/Um (A)0 to Um+1 (A)/Um+1 (A)0 , where Um (A)0 is the connected component of the identity. The K1 -group K1 (A) is defined to be the inductive
limit of the system {Um (A)/Um (A)0 }∞
m=1 . We denote by [u]1 the element in
K1 (A) given by u ∈ Um (A). The quadruple (K0 (A), K0 (A)+ , [1]0 , K1 (A)) is an
isomorphism invariant of A.
If a projection p in a C ∗ -algebra is equivalent to its proper subprojection,
it is said to be infinite. If p is not infinite, it is said to be finite. If every
projection of A is finite, the C ∗ -algebra A is said to be finite. If Mn (A) is finite
for every natural number n, the C ∗ -algebra A is said to be stably finite. For
a stably finite A, the pair (K0 (A), K0 (A)+ ) is an ordered group, that is, we
have K0 (A)+ ∩ (−K0 (A)+ ) = {0}. II1 factors and the UHF-algebras are typical
examples of stably finite C ∗ -algebras.
Let A be a C ∗ algebra not isomorphic to the complex numbers C. If for
every a ∈ A \ {0}, there exist x, y ∈ A satisfying xay = 1, the C ∗ -algebra
A is said to be purely infinite. For a purely infinite C ∗ -algebra A, we have
K0 (A) = K0 (A)+ , and one can drop K0 (A)+ from the above isomorphism
invariant. Type III factors are examples of purely infinite C ∗ -algebras.
The Cuntz algebra On , for n ≥ 2, is the universal C ∗ -algebra
Pngenerated by
isometries S1 , S2 , · · · , Sn with the relations Si∗ Sj = δi,j and i=1 Si Si∗ = 1.
When n = ∞, we define O∞ in a similar way imposing only the first relation.
The Cuntz algebras are separable, nuclear, and purely infinite. For the K-theory
of the Cuntz algebras, we have (K0 (On ), [1]0 , K1 (On )) ∼
= (Z/(n−1)Z, 1, {0}) for
finite n, and (K0 (O∞ ), [1]0 , K1 (O∞ )) ∼
= (Z, 1, {0}).
Kasparov’s KK-theory is a functor associating two abelian groups
KK 0 (A, B) and KK 1 (A, B) to C ∗ -algebras A and B, which are contravariant for A and covariant for B. The group KK 0 (A, B) is often simply denoted
by KK(A, B). We have KK ∗ (C, B) = K∗ (B), and KK ∗ (A, C) = K ∗ (A), the
K-homology group of A. Every homomorphism ρ from A to B gives a KKclass KK(ρ) ∈ KK(A, B). The most remarkable feature of KK-theory is the
existence of an associated product, called the Kasparov product, which is a
generalization of the composition of homomorphisms:
KK i (A, B) × KK j (B, C) 3 (x, y) 7→ x#y ∈ KK i+j (A, C),
where KK ∗+2 (A, B) = KK ∗ (A, B). When there exist x ∈ KK(A, B) and
y ∈ KK(B, A) satisfying x#y = KK(idA ) and y#x = KK(idB ), the two
C ∗ -algebras A and B are said to be KK-equivalent.
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Masaki Izumi
A C ∗ -algebra that is KK-equivalent to an abelian C ∗ -algebra is said to be
K-abelian. We denote by N the category of separable K-abelian C ∗ -algebras,
and call N the UCT class. It is an important open problem whether every
nuclear C ∗ -algebra belongs to N . We denote by Nnuc the set of nuclear C ∗ algebras in N . The UHF-algebras and the Cuntz algebras are in Nnuc .
3.2. Classifiable C ∗ -algebras. In the early stage, the classification of
nuclear C ∗ -algebras developed by extending the classes of building blocks of
inductive limit C ∗ -algebras. When a C ∗ -algebra A has an increasing sequence
of C ∗ -subalgebras {Ak }∞
k=1 whose union is dense in A, we say that A is the
inductive limit of {Ak }∞
k=1 . If each Ak is a direct sum of Ak,j , 1 ≤ j ≤ mk ,
we say that Ak,j is a building block of A. There are several important classes
of inductive limit C ∗ -algebras specified by the classes of building blocks. For
example, if every building block is a matrix algebra, we say that A is an AFalgebra. The UHF-algebras are inductive limits of full matrix algebras, and they
form a subclass of AF-algebras. More generally, if every building block is of the
form pMn (C(Ω))p, where Ω is a compact Hausdorff space and p is a projection
in Mn (C(Ω)), the C ∗ -algebra A is said to be an AH-algebra. AT-algebras, for
which Ω is the circle T = {z ∈ C |z| = 1}, form a subclass of AH-algebras. If
every building block is a subalgebra of Mn (C(Ω)), the C ∗ -algebra A is said to
be an ASH-algebra.
After Glimm’s classification of the UHF-algebras, Bratteli and Elliott classified AF-algebras in the 70s. In modern terms, Elliott’s classification invariant is
the triple (K0 (A), K0 (A)+ , [1]0 ) (the K1 -group is trivial for an AF-algebra). In
the early 90s, Elliott generalized this result to AT-algebras of real rank 0, where
the real rank of a C ∗ -algebra is a generalization of the covering dimension of a
topological space. This is the breakthrough of the remarkable developments of
the classification of nuclear C ∗ -algebras in these two decades.
The first classifiable class of nuclear C ∗ -algebras without referring to inductive limits was discovered in the purely infinite case. A separable simple
nuclear purely infinite C ∗ -algebra is said to be a Kirchberg algebra. The Cuntz
algebras are fundamental examples of Kirchberg algebras. Kirchberg obtained
the following result in the mid 90s.
Theorem 3.1. Let A be a unital nuclear separable simple C ∗ -algebra. Then
the following hold:
(1) The tensor product A ⊗ O2 is isomorphic to O2 .
(2) If A is purely infinite, the tensor product A ⊗ O∞ is isomorphic to A.
The above theorem shows that O2 plays the role of a zero element, and O∞
plays the role of a unit element for tensor product. This fits well with the facts
that O2 is KK-equivalent to {0}, and that O∞ is KK-equivalent to C.
Based on Theorem 3.1, Kirchberg and Phillips showed that the KK-theory
of Kirchberg algebras is given by the asymptotically unitary equivalence classes
of homomorphisms, and they obtained the following classification theorem.
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Theorem 3.2. Let A and B be unital Kirchberg algebras.
(1) The two C ∗ -algebras A and B are KK-equivalent if and only if they are
stably isomorphic, that is, A ⊗ K ∼
= B ⊗ K.
(2) Assume that A, B ∈ Nnuc . Then A and B are isomorphic if and only if
(K0 (A), [1A ]0 , K1 (A)) ∼
= (K0 (B), [1B ]0 , K1 (B)).
For any countable abelian groups M0 and M1 , and any m ∈ M0 , there exists
a model of A satisfying the assumption of (2) such that
(M0 , m, M1 ) ∼
= (K0 (A), [1A ]0 , K1 (A)).
The Elliott program in the stably finite case is still in progress. The first
classifiable class of stably finite C ∗ -algebras without referring to inductive limits
was provided by Huaxin Lin. A unital simple C ∗ -algebra A is said to have tracial
topological rank 0 if it satisfies the following condition: for every ε > 0, for every
a ∈ A+ \ {0}, and for every finite set F ⊂ A, there exists a non-zero projection
p ∈ A satisfying the following:
(1) For every x ∈ F , we have kpx − xpk < ε.
(2) The projection 1 − p is equivalent to a projection in the closure of aAa.
(3) There exists a finite dimensional C ∗ -subalgebra B of pAp and its finite
subset G such that the distance between G and {pxp ∈ A; x ∈ F } is less
than ε.
When one expresses x ∈ F as a matrix
(1 − p)x(1 − p)
x=
px(1 − p)
(1 − p)xp
pxp
,
the condition (1) means that the off diagonal entries are small in norm, and the
condition (2) means that the left up corner is small in the order of projections. If
p can be taken to be 1, the condition (3) is nothing but the local characterization
of AF-algebras (see [8]). Simple AF-algebras and simple AT-algebras of real
rank 0 have tracial topological rank 0.
Lin [29] proved the following classification theorem.
Theorem 3.3. Let A and B be unital simple C ∗ -algebras in Nnuc with tracial
topological rank 0. Then A and B are isomorphic if and only if
(K0 (A), K0 (A)+ , [1A ]0 , K1 (A)) ∼
= (K0 (B), K0 (B)+ , [1B ]0 , K1 (B)).
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Masaki Izumi
An abstract characterization of the quadruple (K0 (A), K0 (A)+ , [1A ]0 ,
K1 (A)) in Theorem 3.3 is also known. A similar result holds in the case of
tracial topological rank 1, where the set of traces are included in the classification invariant (see [30]).
Jiang and Su [20] constructed a unital simple nuclear ASH-algebra Z without nontrivial projections that is KK-equivalent to the complex numbers C.
One can regard Z as a stably finite version of O∞ . Every C ∗ -algebra in known
classifiable classes absorbs Z by tensor product, which is believed to be the key
property for the classification of nuclear C ∗ -algebras. For the latest classification results along this line, the reader is referred to Lin [31], Lin and Niu [32],
and Winter [57].
3.3. The Rohlin property. The Rohlin property, originally coming
from ergodic theory, was first used in operator algebras in Connes’s classification
of cyclic group actions of injective type II factors. In the 90s, its importance
drew attention of specialists in C ∗ -algebras, and systematic analysis of it was
launched. The reader is referred to [12] for the developments of the subject up
to 2000, mainly due to Kishimoto.
We say that an automorphism α of a unital C ∗ -algebra A has the Rohlin
property if for any natural number n, any ε > 0, and any finite set F ⊂ A,
n−1
there exists a partition of unity consisting of projections {ei }i=0
∪ {fi }ni=0 ⊂ A
satisfying
kα(ei ) − ei+1 k < ε,
0 ≤ ∀i ≤ n − 2,
kα(fi ) − fi+1 k < ε,
0 ≤ ∀i ≤ n − 1,
kxei − ei xk < ε,
kxfi − fi xk < ε,
0 ≤ ∀i ≤ n − 1, ∀x ∈ F,
0 ≤ ∀i ≤ n, ∀x ∈ F.
In what follows, we often identify a single automorphism with the Z-action
generated by it. An automorphism is said to be aperiodic if the corresponding Z-action is outer. For the classification of automorphisms α of a C ∗ algebra A up to cocycle conjugacy, the most important property is the stability of α, which is a statement of the following type: every unitary u ∈ A
in a certain class can be approximated by vα(v ∗ ) with v in the same class.
For α to have this property, it needs to be outer in a very strong sense.
The Rohlin property is a means to deduce the stability of α (see [12] for
details).
Except for inductive limit actions, the first classification result of group
actions on C ∗ -algebras was obtained by Kishimoto [25]. For a C ∗ -algebra A with
a unique trace τ , and for α ∈ Aut(A), we denote by α the weakly continuous
extension of α to the weak closure of A in the GNS representation for τ . We
say that α is strongly outer if α is outer.
Group Actions on Operator Algebras
1539
Theorem 3.4. Let A be a UHF-algebra, and let α ∈ Aut(A). The following
conditions are equivalent:
(1) α has the Rohlin property.
(2) αn is strongly outer for any n ∈ Z \ {0}.
(3) The crossed product A oα Z has a unique trace.
(4) The crossed product A oα Z is a simple AT-algebra of real rank 0.
Moreover, there exists a unique cocycle conjugacy class of automorphisms with
the Rohlin property.
The condition (2) is a useful criterion to see if a given automorphism has
the Rohlin property. In fact, it is easy to construct an aperiodic automorphism
without satisfying this condition. The condition (4) means that A oα Z is still
in a classifiable class, which suggests that sufficiently large classifiable classes
should have a permanence property under crossed products by Z-actions with
the Rohlin property. Several authors ([26], [27], [37], [38], [43]) have generalized
various aspects of Theorem 3.4 to simple AT-algebras of real rank 0 and more
generally, simple C ∗ -algebras of tracial rank 0. For the Jiang-Su algebra, Sato
[48] showed that there exists a unique cocycle conjugacy class of strongly outer
Z-actions.
Following Kishimoto’s strategy, Nakamura [40] completely classified aperiodic automorphisms of Kirchberg algebras. The Rohlin property is automatic
in this case.
Theorem 3.5. Every aperiodic automorphism of a unital Kirchberg algebra A
has the Rohlin property. Moreover, the following conditions are equivalent for
two aperiodic automorphisms α, β ∈ Aut(A):
(1) KK(α) = KK(β),
(2) there exist γ ∈ Aut(A) and u ∈ U (A) satisfying KK(γ) = KK(id) and
β = Ad u ◦ γ ◦ α ◦ γ −1 .
To generalize the above results to group actions, we need to formulate the
Rohlin property for group actions first. Nakamura [39] and the author [15]
discussed it for ZN -actions and finite group actions respectively. We present it
in a unified form here.
Let Γ be a countable group. We say that a Γ-action α on a unital C ∗ algebra A has the Rohlin property if for any finite set Γ0 ⊂ Γ \ {e}, there exist
finitely many subgroups Λ1 , Λ2 , . . . , Λr < Γ of finite index such that Γ0 does
not intersect with any conjugate of Λj for 1 ≤ j ≤ r, and the following holds:
for any ε > 0, any finite set F ⊂ A, and any finite set Γ1 ⊂ Γ, there exists a
1540
Masaki Izumi
partition of unity consisting of projections
(j)
(j) αg (ek ) − egk < ε,
(j)
(j) xek − ek x < ε,
Sr
(j)
j=1 {ek }k∈Γ/Λj
⊂ A satisfying
1 ≤ ∀j ≤ r, ∀k ∈ Γ/Λj , ∀g ∈ Γ1 ,
1 ≤ ∀j ≤ r, ∀k ∈ Γ/Λj , ∀x ∈ F.
It is easy to see that this condition forces Γ to be residually finite.
Toward the classification of Γ-actions, we have to take the following two
steps: (1) to show that every (strongly, if A is stably finite) outer Γ-action
has the Rohlin property, (2) to classify Γ-actions with the Rohlin property up
to cocycle conjugacy. When Γ is finite, the Rohlin property is reduced to the
condition with r = 1, Λj = {e}, and it gives a strong K-theoretical constraint.
In fact, there are many K-theoretical obstructions for the step (1), and we
have to give up general classification as in the two theorems above. On the
other hand, the step (2) has already been done by the author. For Γ = ZN , it
seems, at least to the author, that there is no obstruction to these two steps.
We report on the recent progress of these cases in the next two subsections.
3.4. Finite group actions. The reader is referred to [15],[16] for the
proofs of the results stated in this subsection.
Let Γ be a finite group. For a Γ-action α on a C ∗ -algebra A, the Rohlin
property takes the following form: for every ε > 0 and every finite set F ⊂ A,
there exists a partition of unity consisting of projections {eg }g∈Γ in A satisfying
kαg (eh ) − egh k < ε,
kxeg − eg xk < ε,
∀g, h ∈ Γ,
∀g ∈ Γ, ∀x ∈ F.
This condition implies that the following equation holds in K0 (A) if ε is sufficiently small:
X
K0 (αg )([ee ]0 ) = [1]0 ,
g∈Γ
which looks a strong constraint for K0 (A) as a Γ-module. In fact, a much
stronger statement holds. We say that a Γ-module M is cohomologically trivial
b ∗ (Λ, M ) vanishes for every subgroup Λ of Γ (see [4]).
if the Tate cohomology H
If nM is cohomologically trivial for all n ∈ N, we say that M is completely
cohomologically trivial.
Theorem 3.6. Let α be an action of a finite group Γ on a simple unital C ∗ algebra A. If α has the Rohlin property, then K0 (A) and K1 (A) are completely
cohomologically trivial Γ-modules.
This immediately implies that any C ∗ -algebra A with either K0 (A) ∼
= Z or
K1 (A) ∼
= Z, e.g. A = O∞ , has no nontrivial finite group action with the Rohlin
property.
Group Actions on Operator Algebras
1541
Although there is little hope to classify general outer actions of finite groups
on Kirchberg algebras, we have the following theorem for those with the Rohlin
property.
Theorem 3.7. Let Γ be a finite group.
(1) Let A be a unital Kirchberg algebra in Nnuc . If α and β are Γ-actions on
A with the Rohlin property such that K∗ (αg ) = K∗ (βg ) for all g ∈ Γ, then
there exists θ ∈ Aut(A) satisfying K∗ (θ) = 1 and θ ◦ αg ◦ θ−1 = βg for all
g ∈ Γ.
(2) For countable completely cohomologically trivial Γ-modules M0 and M1 ,
there exists a Γ-action α with the Rohlin property on a unital Kirchberg
algebra A in Nnuc such that Ki (A) is isomorphic to Mi as a Γ-module for
i = 0, 1.
The statement (1) holds for A in the class in Theorem 3.3 too.
The Rohlin property for finite group actions is also useful to formulate the
following Γ-equivariant version of Theorem 3.1,(1).
Theorem 3.8. Let α be an outer action of a finite group Γ on a separable simple
unital nuclear C ∗ -algebra A. Then the Γ-action id ⊗α on O2 ⊗ A is conjugate
to a unique (up to conjugate) Γ-action on O2 with the Rohlin property.
If we restrict ourself to Z/2Z-actions on O2 , we have a reasonable classification result. We say that a Z/2Z-action α on a C ∗ -algebra A is strongly
α
approximately inner if there exists a sequence of unitaries {un }∞
n=1 ⊂ A such
∗ ∞
that the sequence {un xun }n=1 converges to α1 (x) for all x ∈ A. If moreover we
can choose un to be self-adjoint, we say that α is approximately representable.
It is easy to see that the dual action of an approximately representable action
has the Rohlin property. Showing that strongly approximate innerness implies
approximate representability in the case of O2 , the author obtained the following theorem by classifying the dual actions.
Theorem 3.9. Let α and β be outer strongly approximately inner Z/2Z-actions
on O2 .
(1) Two actions α and β are cocycle conjugate if and only if their crossed
products are isomorphic.
(2) Two actions α and β are conjugate if and only if their fixed point algebras
are isomorphic.
The K-groups of the crossed product of O2 by any Z/2Z-action are always
uniquely 2-divisible (i.e. multiplying by 2 is a group automorphism). On the
other hand, for any countable uniquely 2-divisible abelian groups M0 , M1 ,
there exists an outer Z/2Z-action α on O2 satisfying Ki (O2 oα Z/2Z) ∼
= Mi for
i = 0, 1. Every known Z/2Z-action on O2 is strongly approximately inner (and
hence approximately representable).
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Masaki Izumi
The reader is referred to [38], [44], [45] for the permanence property of
classifiable classes under the crossed products by finite group actions with the
Rohlin property (or its variant).
3.5. ZN -actions. The Rohlin property for ZN -actions on C ∗ -algebras was
first discussed by Nakamura [39]. He showed that the Rohlin property of Z2 actions on the UHF-algebras is equivalent to strong outerness as in Theorem
3.4, and he classified product type Z2 -actions with the Rohlin property. This
classification result was generalized by Katsura and Matui [24] to general Z2 actions with the Rohlin property on the UHF-algebras (see also [37]).
Matui and the author [18] recently classified a large class of outer Z2 -actions
on a Kirchberg algebra A by KK 1 (A, A). We say that two actions α and β of
a group Γ on A are KK-trivially cocycle conjugate if there exist γ ∈ Aut(A)
with KK(γ) = KK(id) and α-cocycle u satisfying γ ◦ βg ◦ γ −1 = αgu for all
g ∈ Γ.
Theorem 3.10. Let A be a unital Kirchberg algebra.
(1) Every outer ZN -action on A has the Rohlin property.
(2) There exists a one-to-one correspondence between
{x ∈ KK 1 (A, A)| [1]0 #x = 0 ∈ K1 (A)},
and the set of KK-trivially cocycle conjugacy classes of outer Z2 -actions
α with KK(αg ) = KK(id) for all g ∈ Z2 .
We can see from (2) that there exist exactly n − 1 cocycle conjugacy classes
of outer Z2 -actions on the Cuntz algebra On for finite n.
The classification invariant in KK 1 (A, A) arises in the following way. Let
G be a path connected topological group, and let g, h ∈ G with gh = hg. We
choose a continuous path {g(t)}t∈[0,1] in G connecting e and g. Then the path
{g(t)hg(t)−1 h−1 }t∈[0,1] is a loop in G. It is easy to show that the class of the loop
in the fundamental group π1 (G) does not really depend on the choice of the path
{g(t)}t∈[0,1] . For a Z2 -action α, we could apply this argument to G = Aut(A)0 ,
the connected component of id, and the images of the canonical generators
of Z2 if they were in Aut(A)0 . Although our assumption KK(αg ) = KK(id)
does not really imply αg ∈ Aut(A)0 , it is known that θ ∈ Aut(A) satisfies
KK(θ) = KK(id) if and only if θ ⊗ id ∈ Aut(A ⊗ K)0 . Thus we can apply
the argument to the stabilization of α. On the other hand, Dadarlat [6] showed
that π1 (Aut(A ⊗ K)0 ) is isomorphic to KK 1 (A, A), and we get an invariant of
α in KK 1 (A, A).
For ZN -actions, Matui [36], and Matui and the author [18] obtained the
following uniqueness result.
Theorem 3.11. Let A be either O2 , O∞ , or O∞ ⊗ B with B being a UHFalgebra of infinite type (i.e. B ∼
= B ⊗ B). Then there exists a unique cocycle
conjugacy class of outer ZN -actions on A for any natural number N .
Group Actions on Operator Algebras
1543
A unital C ∗ -algebra A 6= C is said to be strongly self-absorbing if there
exists an isomorphism ρ from A onto A ⊗ A that is approximately unitarily
equivalent to the inclusion map A 3 x 7→ x ⊗ 1 ∈ A ⊗ A. A C ∗ -algebra A is said
to be K1 -injective if the canonical map from U (A)/U (A)0 to K1 (A) is injective.
The C ∗ -algebras in the statement of Theorem 3.11 are examples of strongly selfabsorbing K1 -injective C ∗ -algebras. The UHF-algebras of infinite type and the
Jiang-Su algebra Z are other examples. Indeed, Katsura and Matui [24], and
Matui and Sato [38] showed the following.
Theorem 3.12. Let A be either a UHF-algebra of infinite type or the Jiang-Su
algebra Z. Then there exists a unique cocycle conjugacy class of strongly outer
Z2 -actions on A.
3.6. Conjectures. Before ending this note, we clarify what our classification invariant in Theorem 3.10 means in obstruction theory (see [49] for
example), and present two conjectures, which would generalize Theorem 3.10,
Theorem 3.11, and Theorem 3.12.
Let Γ be a discrete group, and let G be a topological group. We denote by
BΓ the classifying space of Γ, and denote by EΓ the universal covering space
of BΓ. To a homomorphism ρ : Γ → G, we can associate a principal G-bundle
Pρ over BΓ, which is the quotient space of EΓ × G by the equivalence relation
(x · γ, g) ∼ (x, ρ(γ)g) for x ∈ EΓ, g ∈ G and γ ∈ Γ. For two homomorphisms
ρ and σ, whether Pρ and Pσ are isomorphic or not can be determined as
follows. Let Iρ,σ be the quotient space of EΓ × G by the equivalence relation
(x · γ, g) ∼ (x, ρ(γ)gσ(γ)−1 ) for x ∈ EΓ, g ∈ G and γ ∈ Γ, which is a fiber
bundle over BΓ. Then the two principal G-bundles Pρ and Pσ over BΓ are
isomorphic if and only if Iρ,σ has a continuous section.
Assume now that Γ = Z2 and G is path connected. For g = ρ((1, 0)) and
h = ρ((0, 1)), the π1 -class of the loop {g(t)hg(t)−1 h−1 }t∈[0,1] discussed in the
previous subsection can be identified with the the primary obstruction class in
H 2 (BZ2 , π1 (G)) = H 2 (T2 , π1 (G)) ∼
= π1 (G),
for the existence of a continuous section of the principal G-bundle Pρ over T2 .
The author would like to thank Sergey Neshveyev for this observation.
Whenever an action α of a group Γ on a C ∗ -algebra A is given, we can
associate to α a principal Aut(A)-bundle Pα over the classifying space BΓ,
where Aut(A) is not necessarily connected. We denote by Aut(A)s the subgroup
of Aut(A ⊗ K) generated by Aut(A) and the inner automorphism group of
A ⊗ K, where we identifying θ ∈ Aut(A) with θ ⊗ id ∈ Aut(A ⊗ K). We regard
the stabilization αs of α as a homomorphism from Γ to Aut(A)s , and denote
by Pαs the corresponding principal Aut(A)s -bundle over BΓ. If two Γ-actions α
and β are cocycle conjugate, their stabilizations are conjugate in Aut(A)s , and
so the two principal Aut(A)s -bundles Pαs and Pβs are isomorphic. When A is a
unital Kirchberg algebra, and KK(αg ) = KK(id) for all g ∈ Γ, we can regard
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Masaki Izumi
αs as a homomorphism from Γ to Aut(A ⊗ K)0 too, and we denote by Pαs,0 the
corresponding principal Aut(A ⊗ K)0 -bundle over BΓ.
Conjecture 1. Let A be a unital Kirchberg algebra, and let Γ be a countable
amenable group whose classifying space BΓ has the homotopy type of a finite
CW complex.
(1) Two outer actions α and β of Γ on A are cocycle conjugate if and only if
the principal Aut(A)s -bundles Pαs and Pβs over BΓ are isomorphic.
(2) Let α and β be outer actions of Γ on A satisfying KK(αg ) = KK(βg ) =
KK(id) for all g ∈ Γ. The two actions α and β are KK-trivially cocycle
conjugate if and only if the principal Aut(A ⊗ K)0 -bundles Pαs,0 and Pβs,0
over BΓ are isomorphic.
Conjecture 1 is true for Γ = Z thanks to Theorem 3.5, and (2) is true for
Γ = Z2 thanks to Theorem 3.10. Finite groups are excluded from the conjecture
because the classifying space BΓ does not have the homotopy type of a finite
CW complex for any nontrivial finite group Γ.
Classical obstruction theory says that whether Pαs,0 and Pβs,0 are isomorphic
or not can be determined by computing relevant cohomology classes in
H n (BΓ, πn−1 (Aut(A ⊗ K)0 )),
2 ≤ n ≤ dim BΓ.
This can be done, at least in principle, because Dadarlat [8] computed the
homotopy groups πn (Aut(A ⊗ K)) for the Kirchberg algebras.
Dadarlat and Winter [7] showed that if A is a strongly self-absorbing K1 injective C ∗ -algebra, the homotopy groups πn (Aut(A)) are trivial for n ≥ 0.
This implies that if α is a Γ-action on such a C ∗ -algebra A, the principal
Aut(A)-bundle Pα over BΓ is trivial.
Conjecture 2. Let Γ be a countable amenable group whose classifying space
BΓ has the homotopy type of a finite CW complex.
(1) If A is either O2 , O∞ or O∞ ⊗ B with B being a UHF-algebra of infinite
type, there exists a unique cocycle conjugacy class of outer Γ-actions on
A.
(2) If A is either a UHF-algebra of infinite type or the Jiang-Su algebra Z,
there exists a unique cocycle conjugacy class of strongly outer Γ-actions
on A.
If Γ is a cocompact lattice of a simply connected solvable Lie group S, we
may choose S for EΓ because S is homeomorphic to Rn . Thus the assumption
on Γ in the two conjectures above is satisfied. For a Γ-action α on a C ∗ -algebra
A, we let
Mα = {f ∈ C b (S, A)| f (xγ) = αγ −1 (f (x)), ∀x ∈ S, γ ∈ Γ},
Group Actions on Operator Algebras
1545
where C b (S, A) is the set of bounded continuous maps from S to A. The C ∗ algebra Mα is identified with the set of continuous sections of the fiber bundle
Pα ×Aut(A) A over BΓ associated with Pα . Therefore the isomorphism class
of Pα determines Mα , and hence the K-theory of Mα . On the other hand,
since the crossed product Mα oλ S by the left translation action λ is stably
isomorphic to the crossed product A oα Γ, the K-theory of Mα is the same
as that of A oα Γ, up to degree change, thanks to repeated use of Connes’s
Thom isomorphism. This means that the isomorphism class of Pα determines
the K-theory of A oα Γ, which is consistent with the two conjectures above.
Acknowledgements
The author would like to thank Hiroki Matui for his critical reading of the first
draft, and Mariko Izumi for everything.
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