THE ROLE OF INVOLUTION IN GRAPH ALGEBRAS
LIA VAŠ
Abstract. Both Leavitt path and graph C ∗ -algebras are equipped with involution. After a
brief introduction to involutive rings, we study the impact of the presence of involution on some
algebraic properties of these two classes of algebras. Whenever possible, we shall point out the
similarities and differences between Leavitt path and graph C ∗ -algebras. We shall also present
a class of open conjectures related to the presence of the involution in these algebras.
Introduction and Overview
Let us start by an overview of your previous lectures, motivation for our study of involution
and an outline of the following lectures.
Question 1. What have we learned from previous lecture? What is a Leavitt path algebra?
A graph C ∗ -algebra?
Answer. E= directed graph (E 0 , E 1 , s, r), K= field (or commutative ring). The Leavitt
path algebra LK (E) is a free K-algebra with basis consisting of vertices, edges, and ghost
edges {e∗ | e ∈ E 1 } with r(e∗ ) = s(e) and s(e∗ ) = r(e) of E such that
(V) vv = v and vw = 0 if v 6= w,
(E1) e = s(e)e = er(e),
(E2) e∗ = r(e)e∗ = e∗ s(e),
(CK1) e∗ e =
and e∗ f = 0 if e 6= f for all e, f ∈ E 1 ,
Pr(e),
(CK2) v = ee∗ for all e ∈ E 1 with v = s(e) and all v ∈ E 0 with 0 < |s−1 (v)| < ∞ (v regular)
for all vertices v and w and all edges e.
Recall that a universal C ∗ -algebra is presented in terms of generators and relations such
that the generators are some bounded operators on a Hilbert space, and the relations imply a
uniform bound on the norm of each generator.
The C∗ -algebra C ∗ (E) is universal C ∗ -algebra generated by {pv |v ∈ E 0 } and {se |e ∈ E 1 }
subject to
(V) p2v = pv = p∗v and pv pw = 0 for v 6= 0
(pv are orthogonal projections),
(PI) se s∗e se = se (se are partial isometries)
(CK1) Same as (CK1) above.
(CK2) Same as (CK2) above.
(CK3) se s∗e ≤ ps(e) (needed for infinite emitters).
Date: July 2015.
2000 Mathematics Subject Classification. 16W99,16W10, 16S99.
Key words and phrases. ∗-ring, Leavitt path algebra, graph C ∗ -algebra, ∗-regular, directly finite, Isomorphism
Conjecture.
1
2
LIA VAŠ
∗
Convince
P ∗yourself∗ that these relations imply (E1). Indeed, er(e) = ee e = e and ∗s(e)e =
ee e + f f e = ee e + 0 = e if s(e) is regular. If s(e) is singular, note that relation se se ≤ ps(e)
implies that ee∗ = s(e)ee∗ . Thus s(e)e = s(e)ee∗ e = ee∗ e = e.
(E2) follows from this by “starring” the relations in (E1).
∗
Question 2. Why are these algebras relevant?
Answer. Graph C ∗ -algebras are examples of many different classes of C ∗ -algebras having a
more convenient and manageable representation than some other algebras in these classes. As a
consequence, some invariants (e.g. K-theory) can be computed easier than for many non-graph
C ∗ -algebras.
Then Leavitt path algebras emerged as algebraic simplification of graph C ∗ -algebras so
that we have the following.
Operator Theory world
↑
Graph C ∗ -algebras
Algebra world
↑
! Leavitt Path Algebras
!
Von Neumann algebras and their algebraic analogue, Baer ∗-rings are in similar relation: one
inhabits operator theory world and the other is its algebraic counterpart. So, the following
quote from the introduction to Berberian’s 1972 book [8], holds for graph C ∗ -algebras as well.
Von Neumann algebras are blessed with an excess of structure – algebraic, geometric,
topological – so much, that one can easily obscure, through proof by overkill, what
makes a particular theorem work.
The algebraic counterpart is needed because of the following.
If all the functional analysis is stripped away ... what remains should (be) completely
accessible through algebraic avenues.
With this in mind, Leavitt path algebras can also be seen as an algebraic avenue paving the
road to understanding its more complex operator theory sibling.
Question 3. What is the goal of these lectures?
Answer. To study one of the “umbrella concepts” common in both worlds – the involution. Both Leavitt path algebras and C ∗ algebras have operation ∗ defined on their elements. So, we are going to look at rings with
involution first and then study the presence of
involution on Leavitt path algebras and graph
C ∗ -algebras.
The consideration of this particular umbrella concept should help each of you find your own
umbrella.
THE ROLE OF INVOLUTION IN GRAPH ALGEBRAS
3
Question 4. What are some open questions or directions of research?
Answer. Characterization of ring theoretic
properties by graph theory, description of
ideals, K-theory, and various generalizations
are just some of the directions of research. We
will be particularly interested in a group of
conjectures known as the Isomorphism Conjectures because they are tied to the involution.
Outline. Below is an outline of the following five lectures.
(1) The driving force. After introducing some basic concepts of rings with involution,
we present some motivating examples for study of ∗-rings. In particular, we look at
examples related to Leavitt path and graph C ∗ -algebras.
(2) We exhibit and study some ring-theoretic properties of involutive rings.
(3) We explore the presence of involution on Leavitt path and graph C ∗ -algebras. In particular, we characterize when a Leavitt path algebra is positive definite and when it is
∗-regular by properties of the underlying graph and the underlying field.
(4) We characterize when a Leavitt path algebra is directly finite.
(5) We consider some open conjectures related to involution referred to as the Isomorphism
Conjectures.
Lecture 1. The Driving Force.
Involution. Let R be an (associative) ring with or without the identity.
Definition 1. R is a ∗-ring (or ring with involution) if there is an operation ∗ : R → R
which is
(1) additive
(x + y)∗ = x∗ + y ∗
(2) anti-multiplicative (xy)∗ = y ∗ x∗
(3) involutive
(x∗ )∗ = x
for all x, y ∈ R.
Let us point out some direct corollaries.
Claim 2. If R is a ∗-ring, then the following hold.
(1) 0∗ = 0.
(2) (−x)∗ = −x∗ for any x ∈ R.
(3) If the ring has the identity 1, then 1∗ = 1.
(1) This follows from additivity since for any x we have that x∗ + 0∗ = (x + 0)∗ = x∗
which implies that 0∗ = 0.
(2) This follows from (1) and additivity since x + (−x) = 0 ⇒ x∗ + (−x)∗ = 0 ⇒ (−x)∗ =
−x∗ .
(3) This follows from anti-multiplicativity since for any x, x∗ 1∗ = (1x)∗ = x∗ and so 1∗ = 1.
Proof.
4
LIA VAŠ
The maps of ∗-rings of interest are not just ring homomorphisms, but ring homomorphisms
f which agree with ∗, i.e. such that
f (x∗ ) = f (x)∗
for every element x. Such maps are called ∗-homomorphisms.
Definition 3. If R is also an algebra over a field (or a commutative ring) K with involution
∗, then R is an ∗-algebra if (ax)∗ = a∗ x∗ for a ∈ K, x ∈ R.
Let us try to think of some examples of involutive rings and algebras. Here are some of
interest.
Example 4.
(1) Any commutative ring is a ∗-ring for ∗ being the identity map. In particular Z, all fields, polynomial rings over fields, etc are ∗-rings.
(2) One of the fields has a special role for C ∗ -algebras – the field of complex numbers C. It
has an involution of special importance – the complex-conjugate involution
a + ib = a − ib.
(3) My favorite non-commutative ring is the ring of matrices. If R is a ring, Mn (R)
denotes the ring of n × n matrices with entries from R. Some special cases of interest
are the following.
(a) The transpose given by (aij )t = (aji ) defines an involution on Mn (R). If R is
commutative, this makes Mn (R) a ∗-algebra with respect to the identity involution
on R.
(b) If R = C with complex-conjugate involution, the map on Mn (C) given by transposing and complex-conjugating each entry ∗ : (aij ) 7→ (aji ) makes Mn (C) a ∗-algebra
over C. This is the only involution considered in C ∗ -algebra world. This example
generalizes to the following.
(c) Most important matrix example. If R = K any field and k 7→ k is an involution
on K, then the map
∗ : (aij ) 7→ (aji )
makes Mn (K) a ∗-algebra over K.
It turns out that this is not just a random example of involution on Mn (K) but
the involution which determines all the other i.e. if 0 is any other involution, then
A0 and A∗ are similar matrices for any matrix A. To see this, note that the algebra
Mn (K) is simple and its center is isomorphic to K. So, by the Skolem–Noether
Theorem, every automorphism on it is inner. This implies the next claim.
Proposition 5. If 0 is any involution on Mn (K) making it a ∗-algebra for involution − on K, then there is an invertible matrix U such that
.
A0 = U A∗ U −1 .
Proof. The composition of involutions ∗ and 0 is an automorphism of Mn (K). By
Skolem–Noether Theorem, there is an invertible matrix V such that the map ∗ ◦ 0
is the same as A 7→ V −1 AV. Thus A0 = (V −1 AV )∗ = V ∗ A∗ (V −1 )∗ for any matrix
A. Take U = V ∗ (which is also invertible with inverse (V −1 )∗ ) and obtain that
A0 = U A∗ U −1 .
Let us also point out a preferred K-basis of Mn (K) – the matrix units eij , i, j =
1, . . . , n. Here eij is the matrix with all entries zero except ij-th which is 1. The
basis elements multiply as follows eij ekl = δjk eil where δjk = 0 if j 6= k and δjj = 1.
THE ROLE OF INVOLUTION IN GRAPH ALGEBRAS
5
(d) The claim analogous to Proposition 5, holds for the transpose and any involution
0
making Mn (K) a ∗-algebra for the identity involution on K. Note also that no
such relation exits for the transpose t and complex-conjugate transpose involution
∗
on Mn (C).
(4) Most relevant example for us. Let K be a field with involution − and E be a
directed graph. The operation ∗, defined on the elements of K and the edges of E,
extends to paths by (e1 . . . en )∗ = e∗n . . . e∗1 and vertices by v ∗ = v. P
It extends to the
entire algebra LK (E) since every element of LK (E) can be written as
apq ∗ where the
sum is finite, a ∈ K and p, q are paths. So, we can define ∗ of such element as follows.
X
∗ X
apq ∗ =
aqp∗ .
Let us explore some specific Leavitt path algebras in the next several examples.
v ↔ e11 w ↔ e22
e / w
• , then LK (E) is ∗-isomorphic to M2 (K) by
.
(5) If E is •v
e ↔ e12 e∗ ↔ e21
You can show that this map induces a ∗-isomorphism by using the universal property of Leavitt path algebras. It states that if R is a K-algebra which contains a set
{av , be , ce∗ |v ∈ E 0 , e ∈ E 1 } such that av , be , ce∗ satisfy axioms (V), (E1), (E2), (CK1),
and (CK2) (such set is called a Leavitt E-family) then there is a unique K-algebra
homomorphism f : LK (E) → R such that f (v) = av , f (e) = be , and f (e∗ ) = ce∗ for all
v ∈ E 0 and e ∈ E 1 . It is direct to show that if a∗v = av∗ and b∗e = ce∗ , then this homomorphism is a ∗-homomorphism. Thus, to show the above map induces a ∗-isomorphism
LK (E) ∼
= M2 (K), it is sufficient to check that the matrix units are a Leavitt E-family.
Similarly, if E is
•v1
e1
/ •v2
e2
/
•vn−1
•
en−1
/ •vn
vi ↔ eii
ei ↔ ei i+1 .
Moreover, if K = C, LC (E) and C ∗ (E) are both ∗-isomorphic to Mn (C).
(6) The algebra of Laurent polynomials over a field K is
P
K[x, x−1 ] = { ni=−m ai xi |m, n nonnegative integers, ai ∈ K}.
If − is an involution on K, the following map is an involution on this algebra.
!∗
n
n
X
X
i
ai x
=
ai x−i
Then LK (E) is ∗-isomorphic to Mn (K) via
i=−m
i=−m
v↔1
If E is •v h e , then LK (E) is ∗-isomorphic to K[x, x−1 ] by
(thus e∗ ↔ x−1 ).
e↔x
(7) Let us turn to an important example of a C ∗ -algebras. In fact, this is the universal
example of a commutative C ∗ -algebra. Let C(X) be the algebra of continuous functions X → C on a compact Hausdorff space X. Addition and multiplication are by
coordinates. The involution is f ∗ (x) = f (x) where − is the complex-conjugate involution. This is an universal example of a commutative C ∗ -algebra since every abelian
C ∗ -algebra is ∗-isomorphic to C(X) for some X.
For example, if E = •v h e , C ∗ (E) is the universal C ∗ -algebra generated by single
unitary operator, i.e. one generator u with relations uu∗ = u∗ u = 1. In this case, C ∗ (E)
is ∗-isomorphic to the algebra of continuous maps on the unit sphere C(S 1 ). The algebra
6
LIA VAŠ
LC (E) embeds in C(S 1 ) by mapping the vertex 1 to the map t 7→ 1 and the edge e to
the map t 7→ eit (thus e∗ is t →
7 e−it ). The algebra C(S 1 ) is also the closure of C[x, x−1 ]
in the norm topology.
e
* w
•
v
(8) Combining examples (5) and (6), let us consider E to be the graph • j
. Then
f
v ↔ e11
e ↔ e12 x
LK (E) ∼
and C ∗ (E) is M2 (C(S 1 )).
= M2 (K[x, x−1 ]) by
f ↔ e21 x
w ↔ e22
e11 ↔ v e11 x ↔ ef
e ↔ f ∗ e12 x ↔ e
Equivalently, one can also do 12
e21 ↔ f e21 x ↔ f ef
e22 ↔ w e22 x ↔ f e
e11 ↔ v
e
↔e
e / w
• h f , then LK (E) ∼
(9) If E is •v
= M2 (K[x, x−1 ]) are ∗-isomorphic by 12
e22 ↔ w
xe11 ↔ f.
(10) Analogously, if E is any finite graph in which no cycle has an exit (i.e. an edge leading
out of any cycle), then every path leads either to a sink or a cycle. In this case, we
have that LK (E) is isomorphic to a direct sum of matrices over K or K[x, x−1 ]. Say
that there are n sinks v1 , . . . , vn and there are ki paths pij , j = 1, . . . , ki , i = 1, . . . , n,
ending in vi and m cycles c1 , . . . , cm and li paths qij , j = 1, . . . , li , i = 1, . . . , m, ending
in a fixed arbitrary vertex of ci , such that qij does not contain ci . Then
m
n
M
M
∼
Mli (K[x, x−1 ])
LK (E) =
Mki (K) ⊕
i=1
i=1
pij p∗il
qij cki qil∗
↔ ejl ∈ Mki (K)
j, l = 1, . . . , ki , i = 1, . . . , n,
−1
k
↔ x ejl ∈ Mli (K[x, x ]) j, l = 1, . . . , li , i = 1, . . . , m, k = 0, 1, . . . .
Practice Problem 1. Identify Leavitt path algebras of the following three graphs as
matrix algebras over K and K[x, x−1 ] or their direct sums.
by
•
•o
•
/
•
•
/
•o
•
•
/
•
/
•O
•
/
•h
(
•
•
(11) Not all Leavitt path algebras have a matricial representation. One of the main examples are the (classical) Leavitt algebras. Consider E to be the graph e 9 • e f .
We shall refer to this graph as the rose with two petals. Recall that the Leavitt
algebra L(1, 2) is the quotient of the free
P K-algebra on four generators x1 , x2 and y1 , y2
subject to relations yi xj = δij 1 and
xi yi = 1. LK (E) is ∗-isomorphic to L(1, 2) by
e ↔ x1 e∗ ↔ y1
The graph C ∗ -algebra of this graph is the Cuntz algebra O2 . This
f ↔ x2 f ∗ ↔ y 2 .
is the universal C ∗ -algebra generated by two generators s1 and s2 subject to relations
s∗i sj = δij 1 for all i, j = 1, 2 and s1 s∗1 + s2 s∗2 = 1.
THE ROLE OF INVOLUTION IN GRAPH ALGEBRAS
e2
The graph
e1
v
6• c
%
•w h
7
also has its Leavitt path algebra ∗-isomorphic to
f2
f1
L(1, 2) and its C ∗ -algebra to O2 . The following induces a ∗-isomorphism between the
v ↔ ee∗ e1 ↔ ee
w ↔ f f ∗ e2 ↔ ef
Leavitt path algebras of the 2-petal rose and of this graph.
f1 ↔ f e
f2 ↔ f f
e3
(12) If E is a rose with n petals
e2
•R re
e1
, LK (E) is the Leavitt algebra L(1, n) and C ∗ (E)
en
is the Cuntz algebra On , the universal C ∗ -algebra
P generated by n generators s1 , . . . , sn
subject to relations s∗i sj = δij 1 for all i, j and i si s∗i = 1.
Combining this with previous examples, LK (E) is ∗-isomorphic to Mm (L(1, n)) for
the graph E below.
e3
•v1
f1
/ •v2
f2
/ • v3
•vm−1
fm−1
/ • vm x
R h
e2
e1
en
(13) If E is a finite graph in which every path leads either to (1) a sink, (2) a cycle without
an exit or (3) a rose with two or more petals, then E is called a polycephaly graph.
The algebra LK (E) is isomorphic to a direct sum of the matrix algebras over (1) K, (2)
K[x, x−1 ] and (3) L(1, n).
(14) Toeplitz algebra is another example of a Leavitt path algebra without a matricial representation. Consider the algebra K[x, y|yx = 1], i.e. the free K-algebra over K with
generators x, y subject to relation yx = 1. This algebra is called the Toeplitz algebra
x∗ = y
over K. If K is an involutive field, we define the involution on this algebra by ∗
.
y =x
If E is the graph •v o
•w h f , then LK (E) is ∗-isomorphic to K[x, y|yx = 1] by the
v ↔ 1 − xy
w ↔ xy
e+f
↔x
map
. In this case we have that ∗
.
e ↔ xxy
e + f∗ ↔ y
f ↔ x − xxy
v ↔ 1 − xx∗
w ↔ xx∗
Equivalently, we can define
and obtain a ∗-isomorphism. The algee ↔ xxx∗
f ↔ x − xxx∗
∗
∗
bra C (E) is the universal C -algebra with one generator x and one relation x∗ x = 1.
(15) We turn now to one of the most important examples of C ∗ -algebras which, in a way,
is an origin of all other involutive rings. Let B(H) be the algebra of bounded operators
on a Hilbert space H. Recall that a Hilbert space is a complex vectors space with an
inner product (a C-valued map on H × H which is linear, adjoint-symmetric, positive,
and faithful) which is complete in the topology given by the norm ||x|| = hx, xi1/2 . For
(x)||
defines the norm
a continuous operator T : H → H, a finite value of supx∈H ||T||x||
e
8
LIA VAŠ
||T ||. This norm makes B(H) into a C ∗ -algebra. The addition of this algebra is by
coordinates, the multiplication is the operator composition, and the involution is given
by the unique map ∗ such that
hT ∗ (x), yi = hx, T (y)i
for any operator T and any x, y ∈ H. If H is finitely dimensional of dimension n, then
B(H) = Mn (C) and the involution is given by the complex-conjugate transpose from
example (3)(b).
We show how our non-matricial algebras examples, Cuntz C ∗ -algebras and Toeplitz
∗
C -algebra can be represented by operators on a Hilbert space.
2
Let ω denote the set of non-negative integers and lP
(ω) denote the Hilbert space of
all sequences of complex numbers ai , i ∈ ω such that i |ai |2 is finite. Both the Cuntz
algebras and the Toeplitz algebra can be represented as operators on this space.
For the Cuntz algebra O2 one can take the generators s1 and s2 to be the following
operators:
s1 : (a0 , a1 , . . .) 7→ (0, a0 , 0, a1 , . . .) and s2 : (a0 , a1 , . . .) 7→ (a0 , 0, a1 , 0 . . .).
∗
Then s1 : (a0 , a1 , . . .) 7→ (a1 , a3 , a5 , . . .) and s∗2 : (a0 , a1 , . . .) 7→ (a0 , a2 , a4 , . . .) so that
e2
s∗i si
= 1, i = 1, 2 and
s1 s∗1 +s2 s∗2
= 1. When considering the graph
e1
6
v
• c
%
•w h
f2
,
f1
v ↔ s1 s∗1
w ↔ s2 s∗2
e1 ↔ s1 s1
e2 ↔ s1 s2
induces a ∗-isomorphism between
we have that the map with
f 1 ↔ s2 s1
f 2 ↔ s2 s2
∗
∗
C (E) and the universal C -algebra with generators s1 , s2 and relations s∗i si = 1, i = 1, 2
and s1 s∗1 + s2 s∗2 = 1.
For Toeplitz algebra, one can take the generator x to be
the unilateral shift x : (a0 , a1 , . . .) 7→ (0, a0 , a1 , . . .).
∗
Then x is the operator (a0 , a1 , . . .) 7→ (a1 , a2 , . . .) so that x∗ x = 1. If π0 denotes the
operator (a0 , a1 , . . .) 7→ (a0 , 0, 0, . . .), then xx∗ = 1 − π0 . So in the graph representation
v ↔ π0
w ↔ 1 − π0
as in example (14), we have the correspondences
.
e ↔ x(1 − π0 )
f ↔ xπ0
(16) Let us conclude our example list with an example of a non-row-finite graph (i.e. there
are infinite emitters) which
Leavitt path algebra is still somewhat related to matrices. If
%
/*4 •w with infinitely countably many edges e1 , e2 , . . . from v to w,
E is the graph •v
then LK (E) is ∗-isomorphic to M∞ (K) ⊕ K1∞ where M∞ (K) stands for matrices with
infinitely countably many rows and columns but with finitely many elements which are
nonzero, and 1∞ stands for the diagonal matrix with infinitely countably many rows and
columns with 1 in all the diagonal entries. The following map induces a ∗-isomorphism.
w ↔ e11
ei ↔ ei+1 1
Note that in this representation ei e∗i ↔ ei+1 i+1 .
v ↔ 1∞ − e11
THE ROLE OF INVOLUTION IN GRAPH ALGEBRAS
9
Positivity, involutions with some nice properties. Having involution, one can define
when an element of a ∗-ring is positive by generalizing the next example.
Example 6. With complex-conjugate involution, for any zj = aj + ibj , j = 1, . . . , n
n
n
X
X
zj zj =
a2j + b2j ≥ 0.
j=1
j=1
Definition 7. An element of a ∗-ring R is positive if it is a finite sum of elements of the form
xx∗ for x ∈ R.
The notation x > 0 usually denotes positive complex numbers. We abuse this notation
slightly and denote the fact that x is positive element by x ≥ 0. If x is positive and nonzero, we
write x > 0. One may argue that we should refer to positive elements as nonnegative instead.
Although this may be a valid point, we continue to use the terminology which is well established
in operator theory and keep referring to such elements as positive.
Definition 8. An involution ∗ on R is
(1) proper if
xx∗ = 0 ⇒ x = 0 for any x ∈ R.
(2) n-proper if
n
X
xi x∗i = 0 ⇒ xi = 0 for each i = 1, . . . , n
i=1
for all x1 , . . . , xn ∈ R.
(3) positive definite if it is n-proper for any n.
Example 9.
(1) The identity involution on R is positive definite.
(2) The identity involution on C is proper but not 2-proper.
(3) The complex-conjugate involution on C is positive definite.
(4) Involution in any C ∗ -algebra is proper. This is because the following.
xx∗ = 0 ⇒ 0 = ||xx∗ || = ||x||2 ⇒ ||x|| = 0 ⇒ x = 0.
(5) Since LC (E) embeds in C ∗ (E) with proper involution for any E, the involution in LC (E)
is proper as well by the previous example.
Questions for the next lecture.
(1) Is the involution of a Leavitt path algebra LK (E) proper for any K?
(2) When is the involution of a Leavitt path algebra positive definite?
We will answer these questions next time. You will be able to make most of the arguments
needed for the answers on your own, if you go over the following practice problems.
Practice Problem 2. Let R be a ∗-ring. Show that the ∗-transpose involution ((aij )∗ = (a∗ji ))
of Mn (R) is proper iff the involution ∗ of R is n-proper. Deduce that every involution of algebras
Mn (R), n = 1, 2, . . . is proper iff the involution ∗ of R is positive definite.
Practice Problem 3. Assuming that the involution on LK (E) is proper for every E, show
that the involution on LK (E) is positive definite for every E. You can assume the following:
adding a line of length n − 1 ending at every vertex of E produces a graph Mn E such that
Mn (LK (E)) ∼
= LK (Mn E).
Practice Problem 4. Recall that a ring is regular if for every a, there is b with a = aba.
Show that R is regular if and only if every principal right (left) ideal is generated by an
idempotent.
10
LIA VAŠ
Lectures 2 and 3. Some ∗-Ring Theoretic Properties of Leavitt Path
Algebras
Let us start with the solutions to the first three practice problems.
Practice Problem 1 Solution. Identify Leavitt path algebras of the following three graphs
as matrix algebras over K and K[x, x−1 ] or their direct sums.
•vO 2
•
c
•v1 o
f
•
e
c
/ •v
2
•
e
/ •v o
f
•
•
e1
/
g
f1
e2
•
f2
/
•
e3
/ •w j
c
+
•w 0
d
•v1
(1) The first graph is a no-exit graph with one sink and one cycle. There are two paths, v1
and e, ending in v1 and two paths, v2 and f, ending in v2 . Thus, LK (E) is ∗-isomorphic
to
M2 (K) ⊕ M2 (K[x, x−1 ]).
(2) The second graph is a no-exit graph with no sinks and a single cycle. There are three
paths v, e and f, ending in v so LK (E) is ∗-isomorphic to
M3 (K[x, x−1 ]).
(3) The third graph is a no-exit graph with two sinks v1 and v2 and one cycle cd. There are
four paths v1 , f2 , f1 f2 , and e1 f2 ending in v1 , and five paths, v2 , g, e2 g, e1 e2 g, and f1 e2 g
ending in v2 . Choosing w for a base of the cycle cd, there are six paths w, e3 , e2 e3 , e1 e2 e3 ,
f1 e2 e3 , and d ending in w. Choosing w0 instead of w for a base of cd would lead to the
same number. Thus, LK (E) is ∗-isomorphic to
M4 (K) ⊕ M5 (K) ⊕ M6 (K[x, x−1 ]).
Practice Problem 2 Solution. Let R be a ∗-ring. Show that the ∗-transpose involution
((aij )∗ = (a∗ji )) of Mn (R) is proper iff the involution ∗ of R is n-proper. Deduce that every
involution of algebras Mn (R), n = 1, 2, . . . is proper iff the involution ∗ of R is positive definite.
Proof. Assume that the involution − is n-proper in R. Suppose A∗ A = 0 for some
Pn matrix A =
∗
(aij ) ∈ Mn (R). Then the diagonal entries of the product A A are zero and so j=1 aij aij = 0
−
for every i = 1, . . . , n. Since
is n-proper, aij = 0 for all i, j. Hence A = 0.
Pn
Conversely, suppose i=1 ai ai = 0 for ai ∈ R. Consider A to be the matrix in Mn (R) that
has the elements a1 , . . . , an in its first row and zeros in the rest of its rows. Then AA∗ = 0.
Since Mn (R) is proper, A = 0. So ai = 0 for every i.
Practice Problem 3 Solution. If the involution on LK (E) is proper for every E, then it is
positive definite for every E.
Proof. By assumptions, ∗ of LK (Mn E) is proper. Thus, ∗ of Mn (LK (E)) is proper and so ∗
of LK (E) is n-proper by Practice Problem 2. Since this holds for every n, we have that ∗ of
LK (E) is positive definite.
This gives us one of the implications in the following theorem.
Theorem 10. ([6, Proposition 2.4]) Let K a field with involution. The following conditions
are equivalent.
THE ROLE OF INVOLUTION IN GRAPH ALGEBRAS
(1) The involution on
(2) The involution on
(3) The involution on
Thus, if E is an arbitrary
definite.
K is positive definite.
LK (E) is positive definite for every graph E.
LK (E) is positive definite for some graph E.
graph, ∗ of LK (E) is positive definite if and only if
11
−
of K is positive
Proof. Showing that (1) implies that ∗ of LK (E) is proper for every E is long and we won’t be
going over that. But, assuming that, we have that ∗ of LK (E) is positive definite by Practice
Problem 3. So, we have (2).
(2) implies (3) is a tautology.
P
(3) ⇒ (1). Suppose that ni=1 ki ki = 0 for ki ∈ K. Let E be a graph such that the involution
P
on
definite. Let v ∈ E 0 . Since v ∗ = v and v 2 = v, 0 = ( ni=1 ki ki )v =
PnLK (E) is positive
∗
0
i=1 (ki v)(ki v) and therefore ki v = 0 for all i by hypothesis. But E is a set of linearly
independent elements in LK (E), so that ki = 0 for all i, as needed.
Idempotents. Idempotent elements of any ring and the following equivalence relation between
them are rather important.
a
e∼
f iff eR ∼
= fR
a
You should convince yourself that ∼
is indeed an equivalence. The following list of equivalent
conditions may come useful.
Practice Problem 5. Part 1. If e and f are idempotents of R, show that the following are
equivalent
a
(1) e ∼
f (recall that we defined this to mean that eR ∼
= f R).
(2) e = xy and f = yx for some x, y ∈ R.
(3) e = xy and f = yx for some x, y ∈ R with ex = x, ye = y, xf = x and f y = y.
(4) Re ∼
= Rf.
Part 2 – Relating the equivalence relations with cancellation properties. Show that for a
unital ring R, idempotent e ∈ R and a R-module P,
a
e∼
1 ⇒ e = 1 if and only if R ⊕ P ∼
= R ⇒ P = 0.
a
Definition 11. If e ∼
f for idempotents e, f of a ring R, then e and f are said to be algebraically equivalent.
a
The idempotent matrices of Mn (R) and relation ∼
on them have special significance because
of the correspondence from the following claim. This correspondence is relevant in order to
better understand the Grothendieck group K0 (R).
Claim 12. Every idempotent matrix E ∈ Mn (R) determines a direct summand of Rn . Conversely, if P is a direct summand of Rn then there is an idempotent matrix E ∈ Mn (R) such
that E(Rn ) is isomorphic to P.
As a consequence, the set of isomorphism classes of finitely generated projective modules is
in bijective correspondence with the set of algebraic equivalence classes on the matrix algebras
Mn (R), n = 1, 2, . . . .
Proof. If E ∈ Mn (R) is an idempotent, then E(Rn ) is a direct summand of Rn since Rn =
E(Rn ) ⊕ (1 − E)(Rn ).
Conversely, if Q is the complement of a direct summand P in Rn , f an isomorphism Rn → P ⊕
Q, πP the projection P ⊕Q → P and iP the inclusion P → P ⊕Q, then the map E = f −1 iP πP f
12
LIA VAŠ
maps Rn into Rn . So E is an element of Mn (R) and its image f −1 (P ⊕0) is isomorphic to P . The
matrix E is an idempotent since E 2 = f −1 iP πP f f −1 iP πP f = f −1 iP πP iP πP f = f −1 iP 1P πP f =
f −1 iP πP f = E.
To prove the second part of the claim, note that if P and T are isomorphic direct summands
of Rn and h is the isomorphism P ∼
= T , then the complement Q of P can be used as the
complement of T as well. If h ⊕ 1Q denotes the map P ⊕ Q → T ⊕ Q given by (p, q) 7→ (h(p), q),
then it is direct to show that iT πT (h ⊕ 1Q ) = (h ⊕ 1Q )iP πP . Thus, if f is an isomorphism
Rn → P ⊕ Q and g an isomorphism Rn → T ⊕ Q, we have that the idempotent matrices
E = f −1 iP πP f and F = g −1 iT πT g are conjugated by the invertible matrix U = g −1 (h ⊕ 1Q )f
a
(i.e. F = U EU −1 ). Thus E ∼
F since E = (EU −1 )U and F = U (EU −1 ).
a
Conversely, if E ∼
F in Mn (R) and if X, Y ∈ Mn (R) are such that XY = E, Y X = F
and X = EX = XF, Y = Y E = F Y (such
elements exists
by Practice Problem 5), then
1−E
X
it is direct to check that the matrix U =
has square equal to the identity
Y
1−F
E 0
0 0
0 1
−1
in M2n (R) and so U
= U. Moreover U
U =
. Since V =
0 0
0 F
1 0
0 0
F 0
E 0
F 0
conjugates
and
, we have that E ⊕ 0 =
and F ⊕ 0 =
0 F
0 0
0 0
0 0
2n ∼
2n
are conjugated and so (E ⊕ 0)(R ) = (F ⊕ 0)(R ). But then we have that
.
E(Rn ) ∼
= (E ⊕ 0)(R2n ) ∼
= (F ⊕ 0)(R2n ) ∼
= F (Rn ).
The next step of our study of idempotents involves right ideals and this requires us to make
a short digression into rings with local units.
Definition 13. A ring R is said to have local units if for every finite set x1 , . . . , xn ∈ R there
is an idempotent u such that xi u = uxi = xi for all i = 1, . . . , n.
Even though some algebras of our interest may not be unital, they all have local units. Let
us recall the related facts in the next example.
Example 14. If the number of vertices of the graph E is finite and v1 , . . . , vn are all the
vertices, then LK (E) is unital algebra and 1 = v1 + . . . + vn .
If the number of vertices of the graph E is not finite, then LK (E) is locally unital algebra.
Indeed, if x1 , . . . , xnP∈ LK (E) and v1 , . . . , vm are the sources of the paths pij , qij in the representation of xi and j aij pij qij∗ , i = 1, . . . , n, then u = v1 + . . . + vm is an idempotent such that
xi u = uxi = xi for all i = 1, . . . , n.
Locally unital rings retain many properties of unital rings. For example, in a non-unital ring
R an element x may not be in the right ideal xR. However, this does not happen in a locally
unital ring since x = xu = ux for some idempotent u and so x = xu ∈ xR.
In the following, let R stand for a ring with local units. The idempotents e and f of a ring
R can be compared as follows.
e ≤ f iff e = f e = ef (iff eR ⊆ f R and Re ⊆ Rf ).
You should convince yourself that ≤ is indeed an order. The equivalence of conditions e = f e
and eR ⊆ f R follows from the following useful observation.
Claim 15. “Useful Observation”. If e is an idempotent of a ring R, then x ∈ eR iff x = ex.
Proof. x ∈ eR ⇒ x = ey for some y. But then ex = eey = ey = x. So, x = ex. The converse
clearly holds.
THE ROLE OF INVOLUTION IN GRAPH ALGEBRAS
13
The condition e = f e is equivalent with e ∈ f R by the previous claim. This last condition is
equivalent with eR ⊆ f R since R has local units.
Projections. Although the conditions eR = f R and Re = Rf together imply that e = f, it is
not the case that the single condition eR = f R implies e = f as it may be desirable. Luckily,
there is a fix if the involution is present. Namely, in ∗-ring, you can work with projections
instead of the idempotents and then the downside does not happen.
Definition 16. An element p of a ∗-ring is a projection iff it is (1) idempotent p2 = p and
(2) self-adjoint p∗ = p.
The order ≤ is nicer for projections than for idempotents since we have the following.
Claim 17. If R is a ∗-ring, the following conditions are equivalent.
1. p ≤ q;
2. p = qp;
3. pR ⊆ qR;
4. Rp ⊆ Rq.
As a consequence,
p = q iff pR = qR.
Proof. 1. trivially implies 2. and 2. trivially implies 3. “Starring” 3. produces 4. Lastly, 4.
implies p = pq and “starring” this relation produces p = qp so 1. holds.
a
The equivalence relation ∼
can also be strengthened in case of projections as follows.
∗
Definition 18. Define a relation ∼
on the set of projections of a ∗-ring R by
∗
p∼
q iff x∗ x = p and xx∗ = q for some x ∈ R.
∗
In this case p and q are said to be ∗-equivalent and the relation p ∼
q is said to be implemented by x.
∗
Claim 19. If p ∼
q is implemented by x, then x can be chosen so that x = xp and qx = x.
Proof. If x∗ x = p and xx∗ = q, let y = xp = xx∗ x = qx and note that y ∗ y = p and yy ∗ = q. Element x from the previous claim is called a partial isometry.1
Claim 20.
(1) An element x of a ∗-ring is a partial isometry iff x = xx∗ x.
(2) If x is a partial isometry, then xx∗ and x∗ x are ∗-equivalent projections with x implementing the ∗-equivalence.
Proof. (1) (⇒) x = xp = xx∗ x. (⇐) If x = xx∗ x, put p = x∗ x and q = xx∗ and check that p
and q are projections with x = xp and qx = x.
(2) If x is a partial isometry, then (xx∗ )(xx∗ ) = (xx∗ x)x∗ = xx∗ and (xx∗ )∗ = (x∗ )∗ x∗ = xx∗
so xx∗ is a projection. Similarly, x∗ x is also a projection. The rest trivially follows.
Example 21. The vertices of a Leavitt path algebra LK (E) are projections and the paths are
partial isometries. Indeed, the vertices are idempotents by axiom (V) and they are selfadjoint
by definition of ∗. If x is a path, then xx∗ x = xr(x) = x so paths are partial isometries.
∗
Claim 22. For every path x of LK (E), r(x) ∼
xx∗ and the equivalence is implemented by x.
Proof. Since r(x) = x∗ x, the claim follows directly from part (2) of Claim 20.
1Recall
that you have encountered both terms “partial isometry” and “projection” in definition of a graph
C ∗ -algebra.
14
LIA VAŠ
This claim can be useful for understanding the monoid of the isomorphism classes of finitely
generated projectives V ((LK (E)) which paves the way to K0 (LK (E)). If E is row-finite, [5,
Theorem 3.5] states that the monoid V ((LK (E)) is isomorphic
P to the monoid ME generated by
the elements av , v ∈ E 0 regular, subject to relations av = e∈s−1 (v) ar(e) . With the knowledge
we accumulated and, in particular, Claim 22, we can understand the arguments for having a
monoid homomorphism
ME → V (LK (E)) given by av 7→ [v]
a
for all vertices v which emit edges where [v] is the ∼
-equivalence class of v. Without going into
the details of the rest of the proof, let us go over just this following argument.
P
Proposition 23. If v is a regular vertex, then v and e∈s−1 (v) r(e) determine the same isomorphism class in V ((LK (E)).
Before the proof, we just need one preparatory claim. Let diag(a1 , . . . , an ) for a1 . . . , an in a
ring R denote the diagonal matrix in Mn (R) with a1 . . . , an on the diagonal.
Claim 24. If p1 , . . . , pn areP
projections in a ∗-ring R with pi pi = 0 for i 6= j (i.e. the projections
∗
are orthogonal), then diag( pi , 0, . . . , 0) ∼
diag(p1 , p2 , . . . , pn ) in Mn (R).
Proof. Convince yourself that the matrix with p1 , p2 , . . . pn in the first row and zeros in all other
rows implements the ∗-equivalence.
Let us prove the proposition now.
Proof. Denote the edges that v emits by ei , i = 1, . . . , n. We have that
P
(by CK2 axiom)
diag(v, 0, . . . , 0) = diag( ei e∗i , 0, . . . , 0)
∗
(by Claim 24)
∼
diag(e1 e∗1 , e2 e∗2 , . . . , en e∗n )
∗
∼ diag(r(e
P 1 ), r(e2 ), . . . , r(en )) (by Claim 22)
∗
∼
diag( r(ei ), 0, . . . , 0).
(by Claim 24)
P
∗
a
a
Since the relation ∼
implies ∼
, we have that diag(v, 0, . . . , 0) ∼
diag( r(ei ), 0, . . . , 0) and
finitely generated projective modules determined by these two matrices are isomorphic by the
proof of 12.
∗-regular rings. Several ring-theoretic properties feature the term “idempotent” in their definitions. For example, Baer, Rickart, and clean rings. We have seen that projections are better
behaved than idempotents. Thus, if the word “idempotent” is replaced by “projection”, in
definitions of such properties for a ∗-ring, one obtains a better behaved property. After recalling Practice Problem 4 from the last lecture, we illustrate this phenomenon by comparing
definitions of regular and ∗-regular rings.
Practice Problem 4 Solution. A ring R is regular if and only if every principal right (left)
ideal is generated by an idempotent.
Proof. If R is regular, a ∈ R and a = aba, then e = ab is an idempotent with aR = eR. Indeed,
bR ⊆ R implies abR ⊆ aR and the converse follows since a = ea ∈ eR and so aR ⊆ eR.
Conversely, if aR = eR, then a = ea and e = ar for some r ∈ R. Then a = ea = ara.
Definition 25. A ring R is regular if and only if every principal right (left) ideal is generated
by an idempotent.
A ∗-ring R is ∗-regular if and only if every principal right (left) ideal is generated by an
idempotent a projection.
THE ROLE OF INVOLUTION IN GRAPH ALGEBRAS
15
The following practice problems relate regular and ∗-regular properties. These practice problems will help us figure out when a Leavitt path algebra is ∗-regular.
Practice Problem 6. Show that if a ∗-ring R is ∗-regular that then it is regular and ∗ is
proper.
Practice Problem 7. Show the converse of the previous claim.
Note that it is sufficient to show that for every idempotent e, there is a projection p such
that eR = pR. The following three claims may be helpful.
(1) If ∗ is proper, then annr (x) = annr (x∗ x) (recall that annr (X) = {y ∈ R | xy = 0 for all
x ∈ X}).
(2) If e is an idempotent, then annr (e) = (1 − e)R.
(3) If e is an idempotent in a (locally) unital ring R, then Re = annl (annr (e)).
(4) If e is an idempotent and R is regular (locally) unital ∗-ring with proper ∗, then Re =
Re∗ e.
Using the above four claims, try to show that for every idempotent e, there is a projection p
such that Re∗ = Rp. “Staring” the previous relation will give you eR = pR.
Now, let us recall the Abrams–Rangaswami characterization of regular Leavitt path algebras.
Theorem 26. ([2, Theorem 1]) LK (E) is regular if and only if E is acyclic.
This theorem also holds if “regular” is replaced by “unit-regular” in the case when E is finite.
We present now the following characterization of the ∗-regular Leavitt path algebras. Let
µ(v) denotes the cardinality of the set of all the paths ending in v.
Theorem 27. ([6, Theorem 3.3]) Let E be an arbitrary graph, K be a field with involution
−
and let σ = sup{µ(v) : v ∈ E 0 } in case the supremum is finite or σ = ω otherwise. The
following conditions are equivalent.
(1) LK (E) is ∗-regular (equivalently, regular and ∗ is proper).
(2) E is acyclic and − of K is n-proper for every finite n ≤ σ.
The equivalence stated in condition (1) follows from Practice Problems 6 and 7. The theorem
follows from Abrams–Rangaswami characterization together with the fact that the involution
−
of K is n-proper for every n ≤ σ if and only if the involution ∗ in LK (E) is proper (note a
certain analogy with Theorem 10).
This result is relevant since it differs from known characterizations of various other algebraic
properties of a Leavitt path algebra LK (E) all of which are given in terms of graph-theoretic
properties of E alone. As opposed to such characterizations, this one involves the underlying
field as well.
We can now prove the following corollary of Theorem 27.
Corollary 28. ([6, Corollary 3.4]) The following conditions are equivalent.
(1) The involution on K is positive definite.
(2) LK (E) is ∗-regular for every acyclic graph E.
Proof. Condition (1) implies that ∗ of LK (E) is positive definite for any graph by Theorem 10.
If E is acyclic, then LK (E) is ∗-regular by Theorem 27 so (2) holds.
Conversely, if (2) holds, let Ln denote a line of length n − 1. Mn (K) ∼
= LK (Ln ) is ∗-regular,
−
its involution is proper, thus of K is n-proper. Since this holds for any n, we have that − of
K is positive definite by Practice Problem 2.
16
LIA VAŠ
Lectures 4 and 5. Finiteness and Direct Finiteness. Isomorphism Conjecture.
Practice Problem 5 Solution. Part 1. If e and f are idempotents of R, show that the
following are equivalent
a
(1) e ∼
f (recall that we defined this to mean that eR ∼
= f R).
(2) e = xy and f = yx for some x, y ∈ R.
(3) e = xy and f = yx for some x, y ∈ R with ex = x, ye = y, xf = x and f y = y.
(4) Re ∼
= Rf.
Part 2. Show that for a unital ring R, idempotent e ∈ R and a R-module P,
a
e∼
1 ⇒ e = 1 if and only if R ⊕ P ∼
= R ⇒ P = 0.
Proof. Part 1. (1) ⇒ (2) Note that a right module isomorphism eR ∼
= f R is determined by
its value at e. If φ is one such isomorphism and φ(e) = y = f y with φ−1 (f ) = x = ex, then
φ(er) = φ(e)r = yr and φ−1 (f r) = xr. Thus, yx = φ(x) = φ(φ−1 (f )) = f and xy = φ−1 (y) =
φ−1 (φ(e)) = e.
(2) ⇒ (3) If x and y are as in (2), then it is direct to check that x0 = exf and y 0 = f ye are
as in (3). In particular, x0 y 0 = exf f ye = exf ye = exyxye = eeeee = e.
(3) ⇒ (1) Define module homomorphisms φ : eR → f R by φ(er) = yr and ψ : f R → eR by
ψ(f r) = xr and check that they are mutually inverse.
(3) ⇒ (4) Define module homomorphisms φ : Re → Rf by φ(re) = rx and ψ : Rf → Re by
ψ(rf ) = ry and check that they are mutually inverse.
(4) ⇒ (2) If φ : Re ∼
= Rf is a left module isomorphism and φ(e) = x = xf with φ−1 (f ) =
y = ye, then φ(re) = rφ(e) = rx and φ−1 (rf ) = ry. Thus, yx = φ(y) = φ(φ−1 (f )) = f and
xy = φ−1 (x) = φ−1 (φ(e)) = e.
Part 2. Let us now show the second part of the problem.
(⇒) Assume that R ⊕ P ∼
= R for some R-module P 6= 0. From relation R ⊕ P ∼
= R
it follows that P is projective since it is a direct summand of the free module R. Let f
denote the isomorphism R → R ⊕ P and let π be the projection of R ⊕ P onto R and i
injection R → R ⊕ 0 ⊆ R ⊕ P. Consider the compositions x = πf and y = f −1 i which are
in HomR (R, R) ∼
= R. Then we have that xy = πf f −1 i = πi = 1R and yx = f −1 iπf. This is
an idempotent since yxyx = f −1 iπf f −1 iπf = f −1 iπiπf = f −1 i1R πf = f −1 iπf = yx. If we
a
denote yx by e. Then we have that xy = 1 and yx = e so e ∼
1. However, e = yx 6= 1 since
for any r ∈ R with f (r) = (s, p) we have that (1 − yx)(r) = r − f −1 iπf (r) = r − f −1 (s, 0) =
r − f −1 ((s, p) − (0, p)) = r − r + f −1 (0, p) = f −1 (0, p). Taking 0 6= p ∈ P, and r = f −1 (1, p) we
obtain that (1 − yx)(r) 6= 0 and so e = yx 6= 1.
(⇐) Conversely, let e ∈ R be an idempotent with xy = e and yx = 1 and we can choose x
such that ex = x and y such that ye = y. Thus eR ∼
= 1R = R and the relations xy = e and
ex = x imply that xR = eR. The relation yx = 1 implies that yR = R. Thus xR ∼
= R and so
R = eR + (1 − e)R = xR ⊕ (1 − xy)R ∼
= R ⊕ (1 − xy)R ⇒ (1 − xy)R = 0.
Since R is unital, this implies xy = 1 so e = 1.
Practice Problem 6 Solution. Show that if a ∗-ring R is ∗-regular that then it is regular
and ∗ is proper.
Proof. If R is ∗-regular then it is also regular because every projection is an idempotent. Now
assume that a∗ a = 0 for some a ∈ R. Then aR = pR for some projection p so a = pa (remember
the “Useful Observation”) and p = ay for some y ∈ R. So, a∗ = a∗ p = a∗ ay = 0. Hence a = 0.
Thus ∗ is proper.
THE ROLE OF INVOLUTION IN GRAPH ALGEBRAS
17
Practice Problem 7 Solution. Show the converse of the previous claim.
Let us show the four claims first.
(1) If ∗ is proper, then annr (x) = annr (x∗ x).
Proof. annr (x) ⊆ annr (x∗ x) is clear since xy = 0 implies x∗ xy = 0. For the converse,
assume that x∗ xy = 0. Left multiply by x∗ and get (xy)∗ xy = y ∗ (x∗ xy) = 0 so that
xy = 0 for any y ∈ R.
(2) If e is an idempotent, then annr (e) = (1 − e)R.
Proof. If ex = 0 then x = x − ex = (1 − e)x ∈ (1 − e)R. Conversely, if x ∈ (1 − e)R,
then x = (1 − e)x (remember the “Useful Observation”) so x = x − ex ⇒ −ex = 0 ⇒
ex = 0 ⇒ x ∈ annr (e).
(3) If e is an idempotent in a (locally) unital ring R, then Re = annl (annr (e)).
Proof. If x ∈ Re, then x = xe. Let y ∈ annr (e). By the second claim y ∈ (1 − e)R, and
so y = (1 − e)y. Thus we have that xy = xe(1 − e)y = 0. This implies that x is a left
annihilator of any y ∈ annr (e). Hence x ∈ annl (annr (e)).
To show the converse, let x ∈ annl (annr (e)) = annl ((1 − e)R), then x(1 − e)y = 0 for
all y ∈ R, that is xy = xey for all y ∈ R. If you assumed that R is unital, this holds for
y = 1 and so x = xe ⇒ x ∈ Re If you assumed that R is a ring with local units, there is
an idempotent u ∈ R such that xu = x and eu = e. Hence x = xu = xeu = xe ∈ Re. (4) If e is an idempotent and R is regular (locally) unital ∗-ring with proper ∗, then Re =
Re∗ e.
Proof. By the regularity of R, Re∗ e = Rf for some idempotent f ∈ R. Thus annr (e∗ e) =
annr (f ) and so annr (e) = annr (e∗ e) = annr (f ) by the first claim (which uses the
assumption that R is proper). So annl (annr (e)) = annl (annr (f )). By the third claim,
this implies that Re = Rf and so Re = Rf = Re∗ e.
Proof. Now let us show the main claim. Assume that R is regular and ∗ is proper. Since every
principal right ideal is generated by an idempotent, it is enough to show that for an arbitrary
idempotent e in R, Re∗ = Rp for some projection p ∈ R. By the fourth claim Re = Re∗ e and
so e = ae∗ e for some a ∈ R. Let p = ae∗ . We claim that p is a projection with Re∗ = Rp. To
see this, note that e = pe and so pp∗ = pea∗ = ea∗ = p∗ . “Starring” the relation p∗ = pp∗ we
obtain that p = (pp∗ )∗ = pp∗ = p∗ . Thus p = p∗ . This also implies that p2 = pp∗ = p∗ = p.
So, p is a projection. The relation p = ae∗ implies that Rp ⊆ Re∗ . The relation e = pe implies
that e∗ = e∗ p and this gives us Re∗ ⊆ Rp. Thus Re∗ = Rp. “Starring” this we have eR = pR.
So, for arbitrary idempotent e, we have found projection p with eR = pR. Regularity and this
fact imply that for any x ∈ R there is a projection p with xR = pR.
Finiteness and direct finiteness. Let us recall the two equivalence relations we introduced.
a
For idempotents: e ∼
f iff e = xy and f = yx for some x and y.
∗
For projections: p ∼ q iff p = xx∗ and q = x∗ x for some x.
If R is unital, the case when f = 1 or q = 1 is of special interest.
Definition 29. A (unital) ring R is said to be directly finite if xy = 1 ⇒ yx = 1 for all
x, y ∈ R.
If R is also a ∗-ring, it is said to be finite if xx∗ = 1 ⇒ x∗ x = 1 for all x ∈ R.
18
LIA VAŠ
When we say that a ∗-ring is finite, this will always mean that the ring is finite in this sense,
not that it has finitely many elements. An operator algebra is a ∗-rings with infinitely many
elements since it contain a copy of C in it. Such algebra being finite means that the identity is
not equivalent to any of its proper subprojections (as the following claim will illustrate). For
von Neumann algebras, this concept correlates exactly to the existence of C-valued dimension
function which takes finite values on all the algebra elements.
a
Claim 30.
(1) A ring is directly finite iff e ∼
1 ⇒ e = 1 for any idempotent e.
∗
(2) A ∗-ring is finite iff p ∼ 1 ⇒ p = 1 for any projection p.
a
∗
The claim relates direct finiteness to a property of ∼
and finiteness to a property of ∼
. It
follows directly from the definitions.
a
By Practice Problem 5, the condition e ∼
1 ⇒ e = 1 for idempotent e is equivalent with
R⊕P ∼
= R ⇒ P = 0 for any module P.
This condition is related to the following cancellation property of the unit-regular rings.
P ⊕T ∼
=Q⊕T ⇒P ∼
= Q for all finitely generated projective modules P, Q, T.
So, the following is a direct corollary of your Practice Problem 5.
Corollary 31. Any unit-regular ring is directly finite.
We present some motivating examples.
Example 32.
(1) Mn (K) is directly finite (since it is unit-regular). If you are doubting
this, recall that Mn (K) can be represented as a Leavitt path algebra of a line of length
n − 1 which is an acyclic graph. Thus Mn (K) is unit-regular by Abrams–Rangaswami
characterization.
(2) Any commutative ring is trivially directly finite. Thus, K[x, x−1 ] is directly finite. Using
Abrams–Rangaswami characterization and the fact that this algebra is isomorphic to
the Leavitt path algebra of •v h e , this ring is not unit-regular. So, K[x, x−1 ] is an
example of directly finite ring which is not unit-regular.
(3) Let E be the graph e 9 • e f . Then e∗ e = 1 and ee∗ + f f ∗ = 1 so if ee∗ were 1, that
would lead us to a contradiction that f f ∗ = 0 ⇒ f = f f ∗ f = 0. So, LK (E) is not finite
and thus not directly finite as well.
(4) The Toeplitz algebra is a universal example of a non-directly finite algebra. Indeed
recall that it can be given as the free K-algebra over K with generators x, y subject to
relation yx = 1. The freeness implies that xy 6= 1 and so this algebra is not directly
finite.
(5) Let V be a countably infinite dimensional vector space over a field K. Consider the
algebra R of all endomorphisms of V. Convince yourself that this algebra can also
be represented as the algebra of matrices with countably many rows and columns and
finitely many nonzero entries in each row. Consider
the matrix unit e11 and let P = e11 R.
v
Then every element of P has the form
where v is a 1 × ∞ matrix with finitely
0
v
v
many nonzero entries and 0 is 0 of R. The map (A,
) 7→
is an isomorphism
0
A
R⊕P ∼
= R. Since P 6= 0, R is not directly finite.
So, some Leavitt path algebras are directly finite while some are not. Our next agenda is the
following.
THE ROLE OF INVOLUTION IN GRAPH ALGEBRAS
19
Agenda. Characterize exactly when a Leavitt path algebra is directly finite.
A necessary condition is relatively easy to get but before we even go there we have a problem.
Problem. What is “1” if the graph is not finite?
To overcome this obstacle, we have to adapt finiteness and direct finiteness to non-unital
rings with local units. Recall that a ring R has local units if for every finite set x1 , . . . , xn ∈ R
there is an idempotent u such that xi u = uxi = xi for all i = 1, . . . , n.
Definition 33. A ring with local units R is said to be directly finite if for every x, y ∈ R
and an idempotent element u ∈ R such that xu = ux = x and yu = uy = y, we have that
xy = u implies yx = u.
A ∗-ring with local units R is said to be finite if for every x ∈ R and an idempotent u ∈ R
such that xu = ux = x, we have that
xx∗ = u implies x∗ x = u.
Condition xx∗ = u implies that u is a projection (selfadjoint idempotent) since u∗ = (xx∗ )∗ =
xx∗ = u. Thus, x∗ u = ux∗ = x∗ as well.
Next on the agenda is to show that these definitions agree with definitions in the unital
case, i.e. that a unital ring R is directly finite in the unital sense iff it is directly finite in the
locally-unital sense. This is taken case of by the following proposition.
Proposition 34. Let R be a unital ring. R is directly finite ring iff xy = e implies yx = e for
any idempotent e and x, y ∈ eRe.
If R is a ∗-ring, R is finite iff xx∗ = p implies x∗ x = p for any projection p and x, y ∈ pRp.
Proof. (⇒) Let xy = e for x, y ∈ eRe. The condition x, y ∈ eRe implies that xe = ex = e
and ye = ey = e and so x(1 − e) = (1 − e)x = 0 and y(1 − e) = (1 − e)y = 0. Then
(1 − e + x)(1 − e + y) = 1 − e + xy = 1 − e + e = 1. The direct finiteness implies that
(1 − e + y)(1 − e + x) = 1 − e + yx = 1. Thus −e + yx = 0 ⇒ yx = e.
(⇐) Take e = 1.
The analogous claim for finiteness is proven similarly.
The algebras in Example 32 indicate that the presence of cycles with exits obstructs direct
finiteness. So, our eyes are on the class of no-exit graphs (recall that we encountered them in
the first lecture).
Definition 35. A graph E is said to be no-exit if s−1 (v) has just one element for every vertex
v of every cycle. If this fails for some cycle, we say that that cycle has an exit.
The next proposition shows that a necessary condition for LK (E) to be (directly) finite is
that E is a no-exit graph.
Proposition 36. ([7, Proposition 3.1], [12, Proposition 4.3]) If E is a graph with a cycle p
which has an exit, then LK (E) is not (directly) finite.
Proof. Let p be a cycle with an exit e. Let v be the source of e and w the range of e.
Case 1. v 6= w. Take x = p + w, and projection u = v + w. Then ux = ux = x and
20
LIA VAŠ
x∗ x = (p∗ + w)(p + w) = v + w = u. However,
xx∗ = (p + w)(p∗ + w) = pp∗ + w. If xx∗ is u,
then pp∗ = v and so e∗ = e∗ v = e∗ pp∗ = 0
which is a contradiction. So, pp∗ 6= v and
xx∗ 6= u. Thus LK (E) is not finite and so it
is not directly finite as well.
Case 2. v = w. Take x = p, u = v. Then
x∗ x = p∗ p = v and xx∗ = pp∗ 6= v similarly as
in Case 1. So we reach the same conclusion.
x
W
p
4
e
•v
x
/ •w
W
p
4
•v h
e
Our goal is to prove that the converse holds as well, in particular, we want to prove the
following.
Theorem 37. ([12, Theorem 4.12]) The following conditions are equivalent.
(1) LK (E) is directly finite.
(2) LK (E) is finite.
(3) E is no-exit.
Note that (1) trivially implies (2) and we have shown that (2) implies (3) in Proposition 36.
Thus, we need to show that (3) implies (1). If E is finite, this is (relatively) manageable given
the matricial representation of a no-exit graph from the first lecture.
Proposition 38. ([1, Theorem 3.7] and [7, Theorem 3.3]) If E is a finite graph, the following
are equivalent.
(1) E is a no-exit graph.
L
L
−1
(2) LK (E) is ∗-isomorphic to ni=1 Mni (K) ⊕ m
i=1 Mmi (K[x, x ]), for some positive integers m, n ni , i = 1, . . . , n, and mi , i = 1, . . . , m.
(3) LK (E) is directly finite.
Proof. The idea of the proof of (1) ⇒ (2) has been sketched during the first lecture. By studying
the properties of matrix algebras, one can obtain (2) ⇒ (3). The implication (3) ⇒ (1) is in
Proposition 36.
Idea of the proof of remaining implication of Theorem 37 for any graph.
(1) Start with x, y in LK (E) for some E no-exit.
(2) Consider a local unit u for x and y such that xy = u. We want to show that yx = u.
(3) Consider a finite subgraph F determined by the paths appearing in x, y, u.
(4) F is a finite no-exit graph and so LK (F ) is directly finite. Thus yx = u. Done.
Problem. LK (F ) may not be a subalgebra of LK (E). So yx = u in LK (F ) does not mean that
yx = u in LK (E) as well. Hence step (4) is problematic.
It turns out that the cure can be found in a concept you may have encountered during Pere
Ara’s lectures – a Cohn-Leavitt algebra. Recall that (CK2) axiom of Leavitt path algebra holds
for the regular vertices (i.e. vertices v which emit nonzero and finitely many edges).
Definition 39. The Cohn-Leavitt algebra CLK (E, S) of a graph E and a subset S of the set
of all regular vertices is obtained in exactly the same way as the Leavitt path algebra except
that the (CK2) axiom holds exactly for those vertices which are in S.
In the case when S is the set of all regular vertices, we obtain the Leavitt path algebra and
we still denote it by LK (E). In the case when S is the empty set, CLK (E, ∅) is call the Cohn
path algebra and it is denoted by CK (E).
THE ROLE OF INVOLUTION IN GRAPH ALGEBRAS
21
Thus we have the following.
Cohn
CK (E)
CK2 holds for
no
regular v’s
Cohn-Leavitt
CLK (E, S)
Leavitt
LK (E)
CK2 holds for
CK2 holds for
some
all
regular v’s
regular v’s
v ∈ S ⇔ CK2 holds
The Cohn-Leavitt algebras also have the C ∗ -counterparts, the relative graph C ∗ -algebras,
considered in [11].
One may think that the class of Cohn-Leavitt algebras is larger than the class of Leavitt path
algebras. This turns out not to be true since every Cohn-Leavitt algebra is in fact ∗-isomorphic
to the Leavitt path algebra of the graph ES defined via E and S (the construction of ES can
be found in [11, Theorem 3.7] and [12, Lemma 4.8]). Thus,
CLK (E, S) ∼
= LK (ES ).
From the construction of graph ES and no-exit characterization for finite graphs (Proposition
38), we have the following.
Proposition 40. ([12, Proposition 4.10]) If E is finite, then CLK (E, S) is (directly) finite if
and only if
(1) E is no-exit and
(2) the vertices of all cycles are in S.
The next result from [4] provides a fix for the problematic step (4) in the idea of our proof of
Theorem 37. Ara and Goodearl prove this result in larger generality for separated graphs.
Proposition 41. ([4, Definition 3.4, Propositions 3.5 and 3.6]) For every finite subgraph G of
a graph E, there are
• a finite subgraph F of E which contains G and
• a subset T of regular vertices of F
such that
CLK (F, T ) is a subalgebra of LK (E).
We can now return to the remaining part of the proof of Theorem 37.
Proof. Recall that it remains to show that condition (3) (E is no-exit) implies condition (1)
(LK (E) is directly finite). Our original idea now works.
Same as in the original idea:
(1) Start with x, y in LK (E) for some E no-exit.
(2) Consider a local unit u for x and y such that xy = u. We want to show that yx = u.
(3) Consider a finite subgraph G determined by the paths appearing in x, y, u.
Different:
(4) Use Proposition 41 to obtain F and T such that CLK (F, T ) is a subalgebra of LK (E).
(5) F is no-exit (since E is) and all the vertices of its cycles are in T by construction.
22
LIA VAŠ
(6) Thus, CLK (F, T ) is directly finite by Proposition 40.
(7) So yx = u in CLK (F, T ) and thus in LK (E) too. Done.
Isomorphism Conjectures. We conclude our exposition on involution in graph algebras by
presenting some open conjectures in which the involution plays a prominent role.
Isomorphism Conjecture (IC). If LC (E) ∼
= LC (F ) as algebras, then LC (E) ∼
= LC (F ) as
∗
∗
∼
∗-algebras (thus C (E) = C (F ) as ∗-algebras by [3, Corollary 4.5]).
Strong Isomorphism Conjecture (SIC). If LC (E) ∼
= LC (F ) as rings, then LC (E) ∼
= LC (F )
∗
∗
∼
as ∗-algebras (thus C (E) = C (F ) as ∗-algebras too).
For these conjectures, it is assumed that the involution on C is fixed to be the complexconjugate involution [3, Definition 2.1]. If the involution is not fixed, the conjecture trivially
fails as the next example shows.
Example 42. Let id denote the identity involution on C and let
denote the complexconjugate involution. Since (C, id) ∼
= (C, ) as fields, L(C,id) (E) ∼
= L(C, ) (E) as algebras.
However, L(C,id) (E) and L(C, ) (E) are not isomorphic as ∗-algebras: if f is a ∗-isomorphism
then i = if (1) = f (i) = f (i∗ ) = f (i)∗ = (f (1)i)∗ = i∗ f (1)∗ = −if (1) = −i. A contradiction.
These conjectures have their “generalized” versions as follows. Let K be a field with (fixed)
involution.
Generalized Isomorphism Conjecture (GIC). If LK (E) ∼
= LK (F ) as algebras, then
LK (E) ∼
= LK (F ) as ∗-algebras.
Generalized Strong Isomorphism Conjecture (GSIC). If LK (E) ∼
= LK (F ) as rings, then
∼
LK (E) = LK (F ) as ∗-algebras.
Generalized Weak Isomorphism Conjecture (GWIC). If LK (E) ∼
= LK (F ) as rings, then
∼
LK (E) = LK (F ) as algebras.
State of the conjectures. [3, Proposition 7.4] shows that (SIC) holds when E and F are
countable acyclic graphs. [3, Proposition 8.5] shows that (SIC) holds if E and F are countable,
row-finite, cofinal (for every vertex v and every infinite path p, there is a vertex w on the path
p such that there is a finite path from v to w) graphs with at least one cycle and such that
every cycle has an exit. Note that the roses with finitely many petals are in this class.
[7, Theorem 6.3] shows that (GSIC) holds for the Leavitt path algebras of finite no-exit
graphs. The proof uses the matricial representation.
In [10] it has been shown that the graded version (GWIC) for finite polycephaly graphs hold.
Some comments on bibliography. Most of the claims on general ∗-rings from these lectures
can be found in [8] and [9]. The papers authored by me are available on arXiv and on my website
at http://www.usciences.edu/˜ lvas.
THE ROLE OF INVOLUTION IN GRAPH ALGEBRAS
23
References
[1] G. Abrams, G. Aranda Pino, F. Perera and M. Siles Molina, Chain conditions for Leavitt path algebras,
Israel J. Math. 165 (2008), 329–348.
[2] G. Abrams, K. Rangaswamy, Regularity conditions for arbitrary Leavitt path algebras, Algebr. Represent.
Theory 13 (3) (2010), 319–334.
[3] G. Abrams, M. Tomforde, Isomorphism and Morita equivalence of graph algebras, Trans. Amer. Math.
Soc., 363 (2011), 3733 – 3767.
[4] P. Ara, K. R. Goodearl, Leavitt path algebras of separated graphs, J. Reine Angew. Math. 669 (2012),
165–224.
[5] P. Ara, M. A. Moreno, E. Pardo, Nonstable K-theory for graph algebras, Algebr. Represent. Theory, 10
(2007), 157–178.
[6] G. Aranda Pino, K. L. Rangaswamy, L. Vaš, ∗-regular Leavitt path algebra of arbitrary graphs, Acta Math.
Sci. Ser. B Engl. Ed. 28 (5) (2012), 957 – 968.
[7] G. Aranda Pino, L. Vaš, Noetherian Leavitt path algebras and their regular algebras, Mediterr. J. Math.,
10 (4) (2013), 1633 – 1656.
[8] S. K. Berberian, Baer ∗-rings, Die Grundlehren der mathematischen Wissenschaften 195, Springer-Verlag,
Berlin-Heidelberg-New York, 1972.
[9] S.K. Berberian, Baer rings and Baer ∗-rings, 1988, https://www.ma.utexas.edu/mp arc/c/03/03-181.pdf.
[10] R. Hazrat, The graded Grothendieck group and classification of Leavitt path algebras, Math. Annalen 355
(2013), no. 1, 273–325.
[11] P. S. Muhly, M. Tomforde, Adding tails to C ∗ -correspondences, Doc. Math. 9 (2004), 79–106.
[12] L. Vaš, Canonical traces and directly finite Leavitt path algebras, Algebr. Represent. Theory (2015), DOI:
10.1007/s10468-014-9513-8.
Department of Mathematics, Physics and Statistics, University of the Sciences, Philadelphia, PA 19104, USA
E-mail address: [email protected]