Leavitt path algebras and graph C ∗ -algebras of
separated graphs (I)
Pere Ara
Universitat Autònoma de Barcelona
July 16, 2015
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
P. Ara, K. R. Goodearl, C*-algebras of separated graphs, J. Funct.
Anal. 261 (2011), 2540–2568.
P. Ara, K. R. Goodearl, Leavitt path algebras of separated graphs,
Crelle’s journal, 669 (2012), 165–224.
P. Ara, Purely infinite simple reduced graph C*-algebras of
one-relator graphs, J. Math. Anal. Appl. 393 (2012), 493–508.
P. Ara, R. Exel, T. Katsura, Dynamical systems of type (m, n) and
their C*-algebras, Ergodic Theory and Dyn. Systems 33 (2013),
1291–1325.
P. Ara, R. Exel, Dynamical systems associated to separated graphs,
graph algebras, and paradoxical decompositions, Advances in
Math. 252 (2014), 748804.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Separated graphs: the initial motivation
Leavitt (1962) defined algebras LK (m, n) for 1 ≤ m ≤ n in the
following way:
LK (m, n) is the K -algebra with generators
{Xji , Xji∗ : 1 ≤ j ≤ m, 1 ≤ i ≤ n}
and defining relations:
XX ∗ = Im ,
X ∗ X = In ,
where X = (Xji ).
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Separated graphs
Definition
A separated
graph is a pair (E , C ) where E is a graph,
F
C = v ∈E 0 Cv , and Cv is a partition of s −1 (v ) (into pairwise
disjoint nonempty subsets) for every vertex v :
G
s −1 (v ) =
X.
X ∈Cv
(In case v is a sink, we take Cv to be the empty family of subsets
of s −1 (v ).)
The constructions we introduce revert to existing ones in case
Cv = {s −1 (v )} for each v ∈ E 0 . We refer to a non-separated
graph in that situation.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
The Leavitt path algebra of a separated graph
Definition
The Leavitt path algebra of the separated graph (E , C ) with
coefficients in the field K , is the K -algebra LK (E , C ) with
generators {v , e, e ∗ | v ∈ E 0 , e ∈ E 1 }, subject to the following
relations:
(V) vv 0 = δv ,v 0 v for all v , v 0 ∈ E 0 ,
(E1) s(e)e = er (e) = e for all e ∈ E 1 ,
(E2) r (e)e ∗ = e ∗ s(e) = e ∗ for all e ∈ E 1 ,
(SCK1) e ∗ e 0 = δe,e 0 r (e) for all e, e 0 ∈ X , X ∈ C , and
P
(SCK2) v = e∈X ee ∗ for every finite set X ∈ Cv , v ∈ E 0 .
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Example
Let 1 ≤ m ≤ n. Let us consider the separated graph
(E (m, n), C (m, n)), where E (m, n) is the graph consisting of two
vertices v , w and with
E (m, n)1 = {α1 , . . . , αn , β1 , . . . , βm },
with s(αi ) = s(βj ) = v and r (αi ) = r (βj ) = w for all i, j, and
C (m, n) consists of two elements X = {α1 , . . . , αn } and
Y = {β1 , . . . , βm }.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Lemma (E. Pardo)
There is a natural isomorphism
γ : LK (m, n) → wLK (E (m, n), C (m, n))w
given by
γ(Xji ) = βj∗ αi ,
γ(Xji∗ ) = αi∗ βj .
This induces an isomorphism
LK (E (m, n), C (m, n)) ∼
= Mm+1 (LK (m, n)).
= Mn+1 (LK (m, n)) ∼
Note that
n
n
X
X
γ(
Xji Xki∗ ) =
βj∗ αi αi∗ βk = βj∗ βk = δjk w
i=1
i=1
∗
j=1 Xji Xjk ) = δik w
Pm
and similarly γ(
so γ is a well-defined
homomorphism, which is shown to be an isomorphism.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
The fact that
LK (E (m, n), C (m, n)) ∼
= Mn+1 (LK (m, n)) ∼
= Mm+1 (LK (m, n))
follows from v ∼ n · w ∼ m · w , since
v=
n
X
αi αi∗ ∼ n · w
i=1
and similarly for the other relation v ∼ m · w .
Therefore
1 = v ⊕ w ∼ (n + 1) · w ∼ (m + 1) · w
Since wLK (E (m, n), C (m, n))w ∼
= LK (m, n), we get the result.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
The Monoid of Projective Modules
Definition
Let R be a ring. The monoid of projective modules, V(R), is the
set of isomorphism classes of finitely generated projective left
R-modules. We endow V(R) with the structure of an abelian
monoid by imposing the operation [P] + [Q] = [P ⊕ Q].
Definition
Let R be a unital ring. The Grothendieck group, K0 (R) is the
universal group of the abelian monoid V(R).
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Definition
If R is not unital, one defines
K0 (R) = Ker(K0 (π) : K0 (R 1 ) → K0 (Z))
where R 1 = R ⊕ Z is the unitization of R.
S
If R has local units (i.e., R = e=e 2 ∈R eRe), then K0 (R) is also
G (V(R)). This applies to LK (E , C ) for all (E , C ).
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Definition
(E , C ) is finitely separated in case |X | < ∞ for all X ∈ C .
Definition
Let (E , C ) be a finitely separated graph. The monoid of (E , C ) is
the abelian monoid M(E , C ) with generators {av | v ∈ E 0 } and
relations
X
av =
ar (e) ,
∀X ∈ Cv , ∀v ∈ E 0 .
e∈X
Theorem (Goodearl-A)
If (E , C ) is a finitely separated graph then the natural map
M(E , C ) → V(LK (E , C ))
is an isomorphism.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Example
For (E , C ) = (E (m, n), C (m, n)), we have
V(L(E , C )) ∼
= M(E , C ) ∼
= ha | ma = nai.
a result originally due to Bergman.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Proposition
If M is any conical abelian monoid, then there exists a bipartite,
finitely separated graph (E , C ) such that
M∼
= M(E , C ) ∼
= V(LK (E , C )).
E can be taken finite if M is finitely generated.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
We remark that, in contrast, the monoids ME ∼
= V(LK (E )) of a
Leavitt path algebra have very special properties:
• ME is conical x + y = 0 =⇒ x = y = 0 (this is a general
property of V(R) for any ring R)
• ME has the Riesz refinement property: If a + b = c + d then
∃x, y , z, t such that a = x + y , b = z + t, c = x + z and d = y + t:
a
b
c
x
z
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
d
y
t
Universitat Autònoma de Barcelona
• ME is a separative monoid: If a + c = b + c and c ≤ na,
c ≤ mb for some n, m ∈ N, then a = b.
where, for x, y in an abelian monoid M, we write x ≤ y in case
y = x + z for some z ∈ M.
• ME is unperforated: na ≤ nb =⇒ a ≤ b.
This was proved by A-Moreno-Pardo.
Even amongst the abelian monoids satisfying all these conditions, the
ones of the form ME are special! (by work of A-Perera-Wehrung)
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Computation of K0
Let (E , C ) be a finitely separated graph. We denote by
0
0
1C : Z(C ) → Z(E ) and At(E ,C ) : Z(C ) → Z(E ) the homomorphisms
defined by
1C (δX ) = δv
if X ∈ Cv
and
At(E ,C ) (δX ) =
X
aX (v , w )δw
(v ∈ E 0 , X ∈ Cv ),
w ∈E 0
where (δX )X ∈C denotes the canonical basis of Z(C ) , (δw ) the
0
canonical basis of Z(E ) and, for X ∈ Cv , aX (v , w ) is the number
of arrows in X from v to w .
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
The next theorem follows from the computation of V(LK (E , C )).
Theorem
Let (E , C ) be a finitely separated graph. Then
0 K0 (LK (E , C )) ∼
= coker 1C − At(E ,C ) : Z(C ) −→ Z(E ) .
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Leavitt types
Definition
Let R be a unital ring. R has IBN if nRR ∼
= mRR =⇒ n = m. (In
terms of idempotents n · 1R ∼ m · 1R =⇒ n = m.)
If R does not have IBN, the Leavitt type of R is the pair (h, k)
defined by the following rules:
(1) h is the smallest positive integer such that hR ∼
= (h + i)R for
some positive i.
(2) k is the smallest positive integer such that hR ∼
= (h + k)R.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
For any unital ring R, the type is detected by V(R), namely:
• R has IBN if the submonoid of V(R) generated by [1R ] is free
(∼
= Z+ )
• R has Leavitt type (h, k) if and only if the submonoid of V(R)
generated by a = [1R ] is ha | ha = (h + k)ai.
Observe that the Leavitt type of R is not detected by K0 (R).
Indeed K0 (R) = G (V(R)) = Z/(k) only detects the second
component of the Leavitt type.
Example
If m = n, then LK (n, n) has IBN. If 1 ≤ m < n, then LK (m, n) has
Leavitt type (m, n − m), because V(LK (m, n)) ∼
= ha | ma = nai.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Definition
• An element a in a conical monoid M is said to be finite if
a + b = a =⇒ b = 0, for b ∈ M. Otherwise a is called infinite.
• A nonzero element a in a conical monoid M is said to be
properly infinite if 2a ≤ a.
• A ring R is called directly finite (the term finite is used in the
context of operator algebras) in case xy = 1 =⇒ yx = 1,
equivalently [1R ] is finite in V(R).
• A ring R is called stably finite in case Mn (R) are directly finite
for all n.
• A ring R is properly infinite if [1R ] is properly infinite in V(R).
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
• For 1 < n, LK (1, n) is properly infinite and simple. Indeed, it is
purely infinite: every nonzero right ideal contains an infinite
idempotent.
• For 1 < m < n, R := LK (m, n) is directly finite but not stably
finite. Indeed 1R , 2 · 1R , . . . , (m − 1) · 1R are finite, but m · 1R is
properly infinite.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Quotients of LK (m, n)
LK (m, n) has many quotients. For instance LK (1, n − m + 1) is a
quotient of LK (m, n):
Im−1
0(m−1)×(n−m+1)
X = (Xji ) 7→
01×(m−1) x1 · · · xn−m+1
and
X ∗ 7→
Im−1
01×(m−1)
0(m−1)×(n−m+1)
x1 · · · xn−m+1
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
∗
Universitat Autònoma de Barcelona
Also, LK (m, n) is a proper quotient essentially of itself!
There is a surjective K -algebra homomorphism
ρ : LK (m, n) → vLK (E (m, n), C (m, n))v
such that
ρ(Xji ) = αi βj∗
ρ(Xji∗ ) = βj αi∗
for i = 1, . . . , n and j = 1, . . . , m.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Amalgamated free products
We will consider only finitely separated graphs, that is |X | < ∞ for
all X ∈ C .
Given ι : A0 → Aι , ι ∈ I , the amalgamated free product
A = ∗A0 Aι is an algebra A, together with ρι : Aι → A such that
ρι ◦ ι = ρι0 ◦ ι0 : A0 → A
for all ι, ι0 , and such that given any other ∗-homomorphisms
δι : Aι → B with δι ◦ ι = δι0 ◦ ι0 for all ι, ι0 there is a unique
δ : A → B such that δ ◦ ρι = δι .
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
L
Let (E , C ) be finitely separated graph. Consider A0 = v ∈E 0 Kv ,
AX = LK (EX ), where (EX )0 = E 0 , (EX )1 = X .
We have ι : A0 → AX and also ρX : AX → LK (E , C ), and of course
ρι ◦ ι : A0 → LK (E , C ) is the natural map.
We have:
LK (E , C ) ∼
= ∗A0 AX ,
the amalgamated free product of AX over A0 .
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
For (E , C ) = (E (m, n), C (m, n)), AX ∼
= Mn+1 (K ),
∼
AY = Mm+1 (K ) , and
LK (E (m, n), C (m, n)) ∼
= Mn+1 (K ) ∗K 2 Mm+1 (K ),
where K 2 → Mn+1 (K ) is given by sending (1, 0) to e11 + · · · + enn
and (0, 1) to en+1,n+1 , and similarly for K 2 → Mm+1 (K ).
Combining this with the isomorphism
Mn+1 (LK (m, n)) ∼
= LK (E (m, n), C (m, n))
we get
Mn+1 (LK (m, n)) ∼
= Mn+1 (K ) ∗K 2 Mm+1 (K ).
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Observe first that in case |E 0 | = 1, we get the free product result
LK (E , C ) ∼
= ∗X ∈C LK (1, |X |).
If, in addition, |X | = 1 for all X ∈ C , we get
LK (E , C ) ∼
= K [F]
where F is the free group on E 1 .
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (I)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Leavitt path algebras and graph C ∗ -algebras of
separated graphs (II)
Pere Ara
Universitat Autònoma de Barcelona
July 16, 2015
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
C*-algebras of separated graphs: Some motivation
Larry Brown (1981) studied the C ∗ -algebra Unnc . Concretely the
algebra Unnc was defined as the universal algebra generated by
elements uij , 1 ≤ i, j ≤ n, subject to the relations making (uij ) a
unitary n × n matrix.
The algebra Unnc was later studied by McClanahan, who also
nc .
studied the rectangular case Um,n
McClanahan raised the problem of computing the K -theory of
nc , establishing a conjecture and proving it in several cases. He
Um,n
nc , and showed that this
also constructed a reduced version of Um,n
reduced version is simple.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Separated graphs
Definition
A separated
graph is a pair (E , C ) where E is a graph,
F
C = v ∈E 0 Cv , and Cv is a partition of s −1 (v ) (into pairwise
disjoint nonempty subsets) for every vertex v :
G
s −1 (v ) =
X.
X ∈Cv
(In case v is a sink, we take Cv to be the empty family of subsets
of s −1 (v ).)
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Definition
For any separated graph (E , C ), the (full) graph C*-algebra of the
separated graph (E , C ) is the universal C*-algebra with generators
{v , e | v ∈ E 0 , e ∈ E 1 }, subject to the following relations:
(V) vw = δv ,w v and v = v ∗ for all v , w ∈ E 0 ,
(E) s(e)e = er (e) = e for all e ∈ E 1 ,
(SCK1) e ∗ f = δe,f r (e) for all e, f ∈ X , X ∈ C , and
P
(SCK2) v = e∈X ee ∗ for every finite set X ∈ Cv , v ∈ E 0 .
In case (E , C ) is trivially separated, C ∗ (E , C ) is just the classical
graph C*-algebra C ∗ (E ).
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Example
Assume that (E , C ) is a separated graph and that |E 0 | = 1. Then
we have
C ∗ (E , C ) ∼
= ∗X ∈C O|X | ,
that is, C ∗ (E , C ) is a free product over C of Cuntz algebras of
type (1, |X |), for X ∈ C .
Note that if |X | = 1 for all X ∈ C then
C ∗ (E , C ) ∼
= C ∗ (Fr ),
a full group C*-algebra of the free group Fr on r = |E 1 | generators.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Lemma
There is a natural isomorphism
nc
γ : Um,n
→ wAm,n w
given by
γ(Xji ) = βj∗ αi ,
γ(Xji∗ ) = αi∗ βj
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Amalgamated free products
We will consider only finitely separated graphs, that is |X | < ∞ for
all X ∈ C .
Given ι : A0 → Aι , ι ∈ I , the amalgamated free product
A = ∗A0 Aι is a C ∗ -algebra A, together with ρι : Aι → A such that
ρι ◦ ι = ρι0 ◦ ι0 : A0 → A
for all ι, ι0 , and such that given any other ∗-homomorphisms
δι : Aι → B with δι ◦ ι = δι0 ◦ ι0 for all ι, ι0 there is a unique
δ : A → B such that δ ◦ ρι = δι .
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Let (E , C ) be finitely separated graph. Consider A0 = C0 (E 0 ),
AX = C ∗ (EX ), where (EX )0 = E 0 , (EX )1 = X .
We have ι : A0 → AX and also ρX : AX → C ∗ (E , C ), and of course
ρι ◦ ι : A0 → C ∗ (E , C ) is the natural map.
We have:
C ∗ (E , C ) ∼
= ∗A0 AX ,
the amalgamated free product of AX over A0 .
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Using reduced amalgamated free products (as defined by
∗ (E , C )
Voiculescu), one can define a reduced graph C*-algebra Cred
of the separated graph (E , C ). This uses canonical conditional
expectations ΦX : AX → A0 , which are faithful.
For the separated graph (E (m, n), C (m, n)), the reduced graph
∗ (E (m, n), C (m, n)) is Morita-equivalent to the
C*-algebra Cred
nc considered by McClanahan.
reduced version of Um,n
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Theorem
If E is a (non-separated) graph then the canonical map
∗ (E ) is an isomorphism.
C ∗ (E ) → Cred
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
The examples revisited
Observe first that in case |E 0 | = 1, we get the free product result
C ∗ (E , C ) ∼
= ∗X ∈C O|X | .
For (E (m, n), C (m, n)), AX ∼
= Mn+1 (C), AY ∼
= Mm+1 (C) , and
C ∗ (E (m, n), C (m, n)) ∼
= Mn+1 (C) ∗C2 Mm+1 (C),
where C2 → Mn+1 is given by sending (1, 0) to e11 + · · · + enn and
(0, 1) to en+1,n+1 , and similarly for C2 → Mm+1 .
Combining this with the isomorphism
nc
Mn+1 (Um,n
)∼
= C ∗ (E (m, n), C (m, n))
we get
nc
Mn+1 (Um,n
)∼
= Mn+1 (C) ∗C2 Mm+1 (C).
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
The canonical conditional expectations ΦX and ΦY are easily seen
to correspond to the maps φ : Mn+1 (C) → C2 and
ψ : Mm+1 (C) → C2 given by
n
φ([aij ]) = (
1X
aii , an+1,n+1 ),
n
i=1
m
1 X
ψ([bij ]) = (
bjj , bm+1,m+1 ).
m
j=1
C ∗ -algebra
∗ (E (m, n), C (m, n))
Cred
The
is simple, but the
∗
∗
C -algebra C (E (m, n), C (m, n)) is far from being simple.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
nuclearity/exactness
• The graph C ∗ -algebra C ∗ (E ) is nuclear for all row-finite graphs.
• For any separated graph (E , C ), the reduced C ∗ -algebra
∗ (E , C ) is exact.
Cred
∗ (E , C ) is not nuclear, e.g.
• In general Cred
∗
∗
Cred
(E , C ) = Cred
(F2 ),
where (E , C ) is the separated graph with one vertex v and two
edges e1 , e2 , and C = {{e1 }, {e2 }}.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Open questions
Open Problems
1
When is the reduced graph C*-algebra of a finitely separated
graph simple?
2
When is the full or reduced graph C*-algebra of a finitely
separated graph finite?
3
∗ (E , C ) is infinite. Is it then purely infinite?
Assume that Cred
Does it at least have real rank zero?
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
We have solved the above problems for some separated graphs. In
particular:
Theorem
Let 1 ≤ m < n. Then the reduced graph C*-algebra
∗ (E (m, n), C (m, n)) is purely infinite simple.
Cred
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Conjecture
Let (E , C ) be a finitely separated graph. Let M(E , C ) be the
abelian monoid
with generators {av | v ∈ E 0 } and relations given
P
by av = e∈X as(e) for all v ∈ E 0 and all X ∈ Cv . Then the
natural map
M(E , C ) → V(C ∗ (E , C ))
is an isomorphism.
• Let LC (E , C ) be the dense ∗-subalgebra of C ∗ (E , C ) generated
by the canonical generators of C ∗ (E , C ). Then (Goodearl, A)
M(E , C ) ∼
= V(LC (E , C )).
• The answer is positive in the non-separated case.
• The answer is positive if we look at stable K -theory:
G (M(E , C )) ∼
= K0 (C ∗ (E , C )).
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Graph C*-algebras and dynamics
It is well-known that graph C*-algebras (of ordinary graphs) are
closely related to dynamics. This was first discovered by Cuntz and
Krieger for On and related C*-algebras OA , nowadays known as
Cuntz-Krieger C*-algebras.
In particular On is related to the shift on X = {1, . . . , n}N .
F
Note that X = ni=1 Hi , with X ∼
= Hi for all i.
(Hi = {(i, x2 , x3 , . . . , )}.)
We extend this to the case (m, n), as follows:
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Dynamical systems of type (m,n)
We study pairs of compact Hausdorff topological spaces (X , Y )
such that
n
m
[
[
X =
Hi =
Vj ,
i=1
j=1
where the Hi are pairwise disjoint clopen subsets of X , each of
which is homeomorphic to Y via given homeomorphisms
hi : Y → Hi . Likewise we will assume that the Vi are pairwise
disjoint clopen subsets of X , each of which is homeomorphic to Y
via given homeomorphisms vi : Y → Vi .
Definition
We will refer to the quadruple X , Y , {hi }ni=1 , {vj }m
j=1 as an
(m, n)-dynamical system.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Definition
An (m, n)-dynamical system X u , Y u , {hiu }ni=1 , {vju }m
j=1 is
universal if it satisfies the following condition: given any
(m, n)-dynamical system
X , Y , {hi }ni=1 , {vj }m
j=1 ,
there exists a unique continuous map
G
G
γ:Ω=X
Y → Ωu = X u
Y u,
such that
1
γ(Y ) ⊆ Y u ,
2
γ(X ) ⊆ X u ,
3
γ ◦ hi = hiu ◦ γ,
4
γ ◦ vj = vju ◦ γ.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Example
When m = 1, the universal (1, n) dynamical system consists of
Xu = S
{1, . . . , n}N , Y u = {10 , . . . , n0 }N , a disjoint copy of X u ,
u
X = ni=1 Hi , where
Hi = {(i, x2 , x3 , . . . , ) : xn ∈ {1, . . . , n}},
hi : Y u → X u sends (x10 , x20 , . . . ) to (i, x1 , x2 , . . . ), and
v : Y u → X u sends (x10 , x20 , . . . ) to (x1 , x2 , . . . ).
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
In general, the universal (m, n) dynamical system is related to the
graph C*-algebra Am,n := C ∗ (E (m, n), C (m, n)), as follows:
Definition
Let U be the subset of partial isometries in Am,n given by
U = {α1 , . . . , αn , β1 , . . . , βm }.
We will let Om,n be the quotient of Am,n by the closed two-sided
ideal generated by all elements of the form
xx ∗ x − x,
as x runs in hU ∪ U ∗ i.
It is worth to mention that A1,n = O1,n ∼
= M2 (On ), because
α1 , . . . , αn , β1 is a tame set of partial isometries when m = 1.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Note that there is a partial action θ of
F Fn+m , the free group on
{a1 , . . . , an , b1 , . . . , bm } on Ωu = X u Y u , obtained by sending ai
to hi and bj to vj .
Theorem
There is a natural isomorphism
Om,n ∼
= C (Ωu ) oθ∗ Fn+m ,
where C (Ωu ) oθ∗ Fn+m denotes the crossed product of the
C*-algebra C (Ωu ) by the induced partial action θ∗ of Fn+m .
We now define the reduced Cuntz Krieger algebra
r
Om,n
= C (Ωu ) orθ∗ Fn+m ,
as the corresponding reduced crossed product.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
All the above can be generalized to any finite bipartite separated
graph (E , C ), obtaining C*-algebras O(E , C ) and Or (E , C ), which
are suitable (full/reduced) crossed products of commutative
C*-algebras by partial actions of free groups.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
The algebra Lab
K (E , C )
The theory is very similar in the purely algebraic case. Let (E , C )
be as before. We look at the construction in some detail:
Set U = hE 1 ∪ (E 1 )∗ i, the multiplicative semigroup of LK (E , C )
generated by E 1 ∪ (E 1 )∗ . For u ∈ U set e(u) = uu ∗ (not an
idempotent in general). Write
0
0
Lab
K (E , C ) = LK (E , C )/h[e(u), e(u )] : u, u ∈ Ui.
It can be shown that {e(u) : u ∈ U} is a family of commuting
idempotents in Lab
K (E , C ).
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Let B be the commutative subalgebra of Lab
K (E , C ) generated by
the idempotents e(u), for u ∈ U.
There exists a totally disconnected, metrizable, compact space
Ω(E , C ) such that
B = CK (Ω(E , C )),
where CK (Ω) denotes the algebra of locally constant functions Ω → K .
Moreover there is a partial action α of F = FhE 1 i on B (given
essentially by conjugation) which induces a partial action α∗ by
homeomorphisms of F on Ω(E , C ). Moreover, we show:
Theorem
∼
Lab
K (E , C ) = CK (Ω(E , C )) oα F.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
We can compute precisely the structure of the monoid
V(Lab (E , C )) thanks to the following approximation result:
Theorem (A-Exel)
There exists a sequence of separated graphs {(En , C n )} canonically
associated to (E , C ) such that (E0 , C 0 ) = (E , C ) and
∼
Lab
L (E , C n ) .
K (E , C ) = lim
−
→ K n
Moreover all the connecting maps LK (En , C n ) → LK (En+1 , C n+1 )
are surjective.
Theorem
∼
V(Lab
M(En , C n ).
K (E , C )) = lim
−
→
Moreover the map M(E , C ) = V(LK (E , C )) → V(Lab
K (E , C )) is an
order-embedding.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Paradoxical decompositions
Let G be a group acting on a set X .
E , E 0 ⊆ X are equidecomposable if
E = A1 t A2 t · · · t An ,
E 0 = B1 t B2 t · · · t Bn
and there exist g1 , g2 , . . . , gn ∈ G such that Bi = gi Ai for all
i = 1, . . . , n.
The type semigroup S(X , G ) is defined by using this relation.
Elements of S(X , G ) are finite sums of equidecomposability classes
[E ], for E ⊆ X .
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
A subset E ⊆ X is called paradoxical if E1 t E2 ⊆ E with
E1 ∼G E and E2 ∼G E .
Note that E ⊆ X is paradoxical ⇐⇒ 2[E ] ≤ [E ] in S(X , G ).
The Banach-Tarski Theorem (or Paradox) asserts that the unit ball
B1 is G-paradoxical, where G is the group of all the isometries of
R3 .
The study of this concept led to the notion of amenable group: A
discrete group Γ is amenable if Γ Γ is not paradoxical.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Tarski’s Theorem
Theorem (Tarski)
Let G be a group acting on a set X . Then the following conditions
are equivalent:
1
E is not G -paradoxical, i.e. 2[E ] [E ]
2
There exists a finitely additive G -invariant measure
µ : P(X ) → [0, +∞] such that µ(E ) = 1.
This result gives the transition from the paradoxical
decompositions characterization of amenable groups to other
characterizations, notably the one involving invariant means.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
About the proof
The proof of Tarski’s Theorem is based on the purely semigroup
theoretic result:
Theorem
Let (S, +) be an abelian semigroup and e ∈ S. Then the following
are equivalent:
(a) There exists a semigroup homomorphism µ : S → [0, ∞] such
that µ(e) = 1.
(b) For all n ∈ N, we have (n + 1)e ne.
and the following properties of S(X , G ):
Schröder-Bernstein axiom: a ≤ b and b ≤ a =⇒ a = b.
Cancellation law: ∀n ∈ N, na = nb =⇒ a = b.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
In fact, with these conditions at hand we can easily show that
condition (b) in the Theorem is equivalent to 2e e, or
equivalently
2e ≤ e ⇐⇒ (n + 1)e ≤ ne for some n.
If (n + 1)e ≤ ne then (n + 1)e = ne by Schröder-Bernstein, and then
(n + 1)e = ne =⇒ n(2e) = ne =⇒ 2e = e by the cancellation law.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
There has been recent interest in trying to extend Tarski’s theorem
to a more general context:
Assume that G acts on a set X and let D be a G -invariant
subalgebra of sets of X . Then one can restrict the
G -equidecomposability relation to elements of D, and obtain a
type semigroup S(X , G , D).
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
In recent papers by Rørdam–Sierakowski and Kerr–Nowak, the
following particular case has been considered:
G acts by homeomorphisms on a totally disconnected compact
Hausdorff space X (e.g. the Cantor set) and D is the subalgebra K
of clopen subsets of X .
These authors have raised the question of whether the analogue of
Tarski’s Theorem holds in this context. More precisely:
Is it true that, for E ∈ K, one has that the following are
equivalent?
(1) 2[E ] [E ] in S(X , G , K),
(2) There exists a semigroup homomorphism
µ : S(X , G , K) → [0, ∞] such that µ([E ]) = 1.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
One may ask:
What are the general properties of S(X , G , K)? It is easy to show
that S(X , G , K) has the following properties:
• It is conical x + y = 0 =⇒ x = y = 0
• It has the Riesz refinement property: If a + b = c + d then
∃x, y , z, t such that a = x + y , b = z + t, c = x + z and d = y + t:
a
b
c
x
z
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
d
y
t
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
We prove that these are the only general properties of S(X , G , K):
Theorem
Let M be an arbitrary f.g. conical abelian monoid. Then there
exists a totally disconnected, metrizable compact space X and an
action of a finitely generated free group F on it such that there is
an order-embedding M ,→ S(X , F, K).
For instance, taking M = ha | na = mai for 1 < m < n one obtains
that there is a clopen subset E ⊆ X such that 2[E ] [E ] in
S(X , F, K), but there is no µ : S(X , F, K) → [0, ∞] such that
µ([E ]) = 1.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
In the general setting of a partial action θ of a group Γ on a totally
disconnected compact space X , we always have a monoid
homomorphism:
S(X , Γ, K) −→ V(CK (X ) oθ∗ Γ)
[Y ] 7→ χY · δe
If X = Ω(E , C ) for a finite bipartite separated graph (E , C ), we
are able to show:
Theorem
The natural homomorphism
S(Ω(E , C ), F, K) −→ V(CK (Ω(E , C )) oα F)
is an isomorphism
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Now, starting with a finitely generated conical abelian monoid M,
we choose a finite bipartite separated graph (E , C ) such that
M∼
= M(E , C ), and so we get a totally disconnected metrizable
compact space Ω(E , C ) with a partial action α∗ of F = FhE 1 i such
that there is an order-embedding
M ,→ V(Lab (E , C )) ∼
= S(Ω(E , C ), F, K).
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Finally, using globalization techniques due to Abadie, we can reach
the same conclusion, but with total actions instead of partial
actions, obtaining:
Theorem
Let M be an arbitrary f.g. conical abelian monoid. Then there
exist a totally disconnected, metrizable compact space X and an
action of a finitely generated free group F on it such that there is
an order-embedding M ,→ S(X , F, K).
Corollary
There exist a global action of a finitely generated free group F on a
totally disconnected metrizable compact space Z , and a
non-F-paradoxical (with respect to K) clopen subset A of Z such
that µ(A) = ∞ for every finitely additive F-invariant measure
µ : K → [0, ∞] such that µ(A) > 0.
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona
Examples Amalgamated free products The examples revisited Open questions Graph C*-algebras and dynamics Paradoxical deco
Thank you very much for your attention!!!
Pere Ara
Leavitt path algebras and graph C ∗ -algebras of separated graphs (II)
Universitat Autònoma de Barcelona