Branching Systems and General Cuntz-Krieger Uniqueness Theorem for Ultragraph C*-algebras

0.33 MB·

by  Daniel Gonçalves

Branching Systems and General Cuntz-Krieger
Uniqueness Theorem for Ultragraph
6
1 C*-algebras
0
2

n
a Daniel Gonc¸alves, Hui Li and Danilo Royer
J ∗ †

6
2 26 Jan 2016

]
A
O
AMS 2000 MSC: 47L99, 37A55
.
h
t Keywords and phrases: Ultragraph; ultragraph C*-algebra; branching
a
m system; Perron-Frobenius operator; permutative representation; faithful rep-
[
resentation; general Cuntz-Krieger uniqueness theorem.

1
v
Abstract
5
0
0 We give a notion of branching systems on ultragraphs. From this
7
0 we build concrete representations of ultragraph C*-algebras on the
.
1 boundedlinear operators of Hilbertspaces. To each branchingsystem
0
ofanultragraphwedescribetheassociated Perron-Frobeniusoperator
6
1 in terms of the induced representation. We show that every permu-
:
v
tative representation of an ultragraph C*-algebra is unitary equiva-
i
X
lent to a representation arising from a branching system. We give a
r
a sufficient condition on ultragraphs such that a large class of repre-
sentations of the C*-algebras of these ultragraphs is permutative. To
give a sufficient condition on branching systems so that their induced
representations are faithful we generalize Szyman´ski’s version of the
Cuntz-Krieger uniqueness theorem for ultragraph C*-algebras.
∗Partially supported by CNPq.
†SupportedbyResearchCenterforOperatorAlgebrasofEastChinaNormalUniversity.
The corresponding author.
1
1 Introduction
Ultragraphs are combinatorial objects that generalize directed graphs.
Roughly speaking an ultragraph is a graph where the image of the range
map does not belong to the set of vertices, but instead to the power set
on vertices. The concept was introduced by Mark Tomforde in [24] with
an eye towards C*-algebra applications. In particular, Tomforde showed in
[24] how to associate a C*-algebra to an ultragraph and proved that there
exist ultragraph C*-algebras that are neither Exel-Laca algebras nor graph
C*-algebras. So the study of ultragraph C*-algebras is of interest, but fur-
thermore, ultragraph C*-algebras were key in answering the long-standing
question of whether an Exel-Laca algebra is Morita equivalent to a graph
algebra (see [16]).
Recently the scope of ultragraphs has surpassed the realm of C*-algebras,
reaching applications to symbolic dynamics. In particular, ultragraphs are
fundamental in characterizing one-sided shift spaces over infinite alphabets.
Furthermore, questions regarding the dynamics of one-sided shift spaces over
infinite alphabetswereanswered using ultragraphsandtheir C*-algebras(see
[13]).
The above evidence leads us to believe that there are still many appli-
cations of ultragraphs to be found. Indeed, among the results we present
in this paper, we will show a connection (via branching systems) between
ultragraphs and the Perron-Frobenius operator from the ergodic theory (see
Section 5), therefore generalizing results previously obtained for graph alge-
bras and Cuntz-Krieger C*-algebras (see [10, 12]).
Branching systems arenotjustimportant asaway toconnect ultragraphs
to the ergodic theory. They have appeared in fields as random walks, sym-
bolic dynamics, wavelet theory and are strongly connected to the representa-
tion theory of combinatorial algebras. In particular, Bratteli and Jorgensen
have initiated the study of wavelets and representations of the Cuntz algebra
via branching systems in [3, 4]. After this, many results relating branching
systems and representations of generalized Cuntz algebras were obtained, see
for example [7, 8, 9, 11, 12, 14]. It is our goal inthis paper to generalize many
2
of the results obtained in the just mentioned manuscripts to ultragraphs.
In particular, to obtain ultragraph versions of the results presented in [8]
we must first develop a generalized Cuntz-Krieger uniqueness theorem for
ultragraph C*-algebras. Actually there has been a lot of activity recently
on generalized Cuntz-Krieger theorems for combinatorial algebras, see for
example [5, 6, 26] and so the proof of a generalized Cuntz-Krieger uniqueness
theorem for ultragraph C*-algebras is of independent interest.
We break the paper as follows. In Section 2 we present a preliminary
on ultragraphs. In Sections 3, 4 and 5 we generalize the results in [10] to
ultragraphs. In particular we introduce branching systems of ultragraphs in
Section 3; In Section 4 we show how to build a representation of the ultra-
graphC*-algebrafromabranching systemofanultragraph; theninSection5
we find the connection between branching systems, representations and the
Perron-Frobenius operator. We generalize the results in [14] in Section 6,
where we introduce permutative representations, show that they are always
unitarily equivalent to a representation arising from a branching system and
give sufficient conditions on ultragraphs so that a large class of represen-
tations on these ultragraph C*-algebras are permutative. In Section 7 we
make a pause in the theory of branching systems and prove the generalized
Cuntz-Krieger uniqueness theorem for ultragraph C*-algebras. This section
can be read independently from the other sections. To end the paper, in
Section 8 we prove versions of the results in [8] for ultragraphs, that is, we
prove a converse of the Cuntz-Krieger uniqueness theorem for ultragraphC*-
algebras and give a sufficient condition for a representation of an ultragraph
C*-algebra, arising from a branching system, to be faithful.
2 Preliminaries
Throughoutthispaper, thenotationNstandsforthesetofallthepositive
integers; all measure spaces are assumed to be σ-finite; all C*-algebras are
assumed to be separable; and all representations of C*-algebras are assumed
to be on separable infinite-dimensional Hilbert spaces.
The following lemma might be well-known but we are not able to find
3
any reference to it.
Lemma 2.1 Let A be a C*-algebra, let I,J be closed two-sided ideals of A,
let B be a C*-subalgebra of I, and let X be a closed subspace of I. Suppose
that BX,XI X;XX∗ B,X∗X I; and spanXX∗ = B,spanX∗X = I.
⊂ ⊂ ⊂
Furthermore, suppose that B J = 0. Then I J = 0.
∩ ∩
Proof. It is straightforward to show that X is a B–I imprimitivity bi-
module (see [23, Definition 3.1]) with the natural operations on X. By [23,
Theorem 3.22], there exists a lattice isomorphism Φ from the set of closed
two-sided ideals of B onto the set of closed two-sided ideals of I. A straight-
forward calculation gives
Φ(B J) = span x∗y : x,y X; and for z X, we have yz∗ J B .
∩ { ∈ ∈ ∈ ∩ }
We claim that Φ(B J) = I J. Since X I,Φ(B J) I. Fix x,y X
∩ ∩ ⊂ ∩ ⊂ ∈
satisfying that for z X,yz∗ J B. Notice that x∗y( n z∗z′) J for
∈ ∈ ∩ i=1 i i ∈
z ,z′,...,z ,z′ X. Since spanX∗X = I, we deduce that x∗y J. So
1 1 n n ∈ P ∈
Φ(B J) I J. Conversely, fix n x∗x′ spanX∗X J. Take an
∩ ⊂ ∩ i=1 i i ∈ ∩
approximate identity (e ) I J. Then x′e z∗ J B for all α, for all
α ⊂ ∩ P i α ∈ ∩
z X , and n x∗x′e n x∗x′. So n x∗x′ Φ(B J). Since
∈ i=1 i i α → i=1 i i i=1 i i ∈ ∩
spanX∗X = I,I J Φ(B J). Hence Φ(B J) = I J. Therefore
P ∩ ⊂ P∩ P ∩ ∩
I J = 0 because B J = 0. (cid:3)
∩ ∩
Remark 2.2 Φ as in Lemma 2.1 is indeed the inverse map of the Rieffel
correspondence given in [23, Proposition 3.24].
In the rest of this section we give a brief introduction to ultragraphs and
ultragraph C*-algebras, as defined by Tomforde in [24].
Definition 2.3 ([24, Definition 2.1]) An ultragraph is a quadruple =
G
(G0, 1,r,s) consisting of two countable sets G0, 1, a map s : 1 G0 and
G G G →
a map r : 1 P(G0) , where P(G0) stands for the power set of G0.
G → \{∅}
Example 2.4 Below we show the picture of the ultragraph where G0 =
v ,...v , 1 = e ,...,e and the source and range maps are defined
1 10 1 5
{ } G { }
4
as follows: s(e ) = v , s e ) = v , s(e ) = v , s(e ) = v and s(e ) = v ;
1 1 ( 2 6 3 2 4 6 5 10
r(e ) = v ,v , r(e ) = v ,v ,v , r(e ) = v , r(e ) = v and
1 2 3 2 3 4 5 3 7 4 10
{ } { } { } { }
r(e ) = v ,v .
5 8 9
{ }
e vr2 >e3 rv7 vr8 v9r
v r >1
1 rv
3
v4r <e2 r e>4 ∧re5
v v
6 10
v r
5
Ultragraph .
G
Remark 2.5 Notice how in the drawing above we split the edges to represent
their range.
Definition 2.6 ([24, Page 349]) Let be an ultragraph. Define 0 to be
G G
the smallest subset of P(G0) that contains v for all v G0, contains
{ } ∈
r(e) for all e 1, and is closed under finite unions and nonempty finite
∈ G
intersections.
The set 0 can be characterized in the following way.
G
Lemma 2.7 ([24, Lemma 2.12]) Let be an ultragraph. Then
G
0 = r(e) r(e) F : X ’s are finite subsets of 1,
i
G ∪···∪ ∪ G
n(cid:16)e\∈X1 (cid:17) (cid:16)e∈\Xn (cid:17)
F is a finite subset of G0 .
o
Definition 2.8 ([24, Definition 2.7, Theorem 2.11]) Let be an ultra-
G
graph. The ultragraph algebra C∗( ) is the universal C*-algebra generated
G
by a family of partial isometries with orthogonal ranges s : e 1 and a
e
{ ∈ G }
family of projections p : A 0 satisfying
A
{ ∈ G }
1. p = 0,p p = p ,p = p +p p , for all A,B 0;
∅ A B A∩B A∪B A B A∩B
− ∈ G
2. s∗s = p , for all e 1;
e e r(e) ∈ G
5
3. s s∗ p for all e 1; and
e e ≤ s(e) ∈ G
4. p = s s∗ whenever 0 < s−1(v) < .
v e e | | ∞
s(e)=v
P
Moreover, any family of partial isometries with orthogonal ranges S : e
e
{ ∈
1 and any family of projections P : A 0 in any C*-algebra B satis-
A
G } { ∈ G }
fying Conditions (1)–(4) is called a Cuntz-Krieger -family.
G
Notice that p s s∗ for any nonempty finite subset S s−1(v).
v ≥ e∈S e e ⊂
It follows from [24, Theorem 2.11] that each s and p , with A = , are
P e A 6 ∅
nonzero.
Definition 2.9 ([24, Page 350]) Let be an ultragraph. For α 0, de-
G ∈ G
fine s(α) = r(α) := α, and define α := 0. For n 1, define n := α =
| | ≥ G {
(α )n n 1 : s(α ) r(α ),i = 1,...,n 1 , and for α n define
i i=1 ∈ i=1G i+1 ∈ i − } ∈ G
s(α) := s(α ),r(α) := r(α ), α := n. Define ∗ := ∞ n. For α ∗,
Q1 n | | G ∐n=0G ∈ G
define
s ...s if α > 0
s := α1 α|α| | |
α

p if α = 0.
 α | |

Definition 2.10 ([24, Definition 3.4]) Let be an ultragraph. An ele-
G
ment α ∗ 0 is called a cycle if s(α ) r(α ). A cycle α is said to be
1 n
∈ G \G ∈
simple if α = α for all i = j. A cycle α is said to have exits if one of the
i j
6 6
following happens:
1. there exists 1 i n 1 such that s−1(r(α )) = α ;
i i+1
≤ ≤ − 6
2. s−1(r(α )) = α ;
n 1
6
3. there exist 1 i n and v r(α ) such that s−1(v) = .
i
≤ ≤ ∈ ∅
Moreover, the ultragraph is said to satisfy Condition (L) if every cycle has
G
exits.
It is straightforward to see that a cycle α = (α ,...,α ) has no exits if
1 n
r(α ) = 1,s−1(s(α )) = α for all i = 1,...,n.
i i i
| | { }
6
Definition 2.11 ([25, Definitions 3.1, 3.2]) Let be an ultragraph. A
G
subset 0 is said to be hereditary if
H ⊂ G
1. for A,B , we have A B ;
∈ H ∪ ∈ H
2. for A ,B 0, if B A then B ; and
∈ H ∈ G ⊂ ∈ H
3. for e 1, if s(e) then r(e) .
∈ G { } ∈ H ∈ H
A subset 0 is said to be saturated if for v G0,0 < s−1(v) <
S ⊂ G ∈ | |
,r(s−1(v)) = v .
∞ ⊂ S ⇒ { } ∈ S
Lemma 2.12 ([25, Lemma 3.12]) Let be an ultragraph and let be a
G H
hereditary subset of 0. Define := . For n 0, define
0
G H H ≥
S := v G0 : 0 < s−1(v) < ,r(s−1(v)) ; and define
n n
{ ∈ | | ∞ ⊂ H }
:= A F : A ,F is a finite subset of S .
n+1 n n
H { ∪ ∈ H }
Then ∞ isthe smallesthereditaryandsaturated subsetof 0 containing
n=0Hn G
.
H S
Lemma 2.13 ([25, Lemma 3.5]) Let be an ultragraph and let be
G SH
a hereditary and saturated subset of 0. Then the closed two-sided ideal
G
I( ) of C∗( ) generated by p : A is gauge-invariant and has the
A
SH G { ∈ SH}
following form.
I( ) = span s p s∗ : α,β ∗,A .
SH { α A β ∈ G ∈ SH}
The following is the Cuntz-Krieger uniqueness theorem for ultragraph
C*-algebras.
Theorem 2.14 ([24, Theorem 6.7]) Let be an ultragraph which satis-
G
fies Condition (L). Then for a Cuntz-Krieger -family P ,S : A 0,e
A e
G { ∈ G ∈
1 in a C*-algebra B satisfying P = 0 whenever A = , the ultragraph C*-
A
G } 6 6 ∅
algebra C∗( ) is isomorphic with C∗(P ,S ) via the map p P ,s S .
A e A A e e
G 7→ 7→
7
3 Branching Systems of Ultragraphs
In this section we introduce branching systems associated to ultragraphs
and show that they always exist.
Definition 3.1 Let be an ultragraph, (X,µ) be a measure space and let
G
R ,D be a family of measurable subsets of X. Suppose that
e A e∈G1,A∈G0
{ }
1. R R µ−=a.e. if e = f 1;
e f
∩ ∅ 6 ∈ G
2. D = ;D D µ−=a.e. D ;D D µ−=a.e. D for all A,B 0;
∅ A B A∩B A B A∪B
∅ ∩ ∪ ∈ G
µ−a.e.
3. R D for all e 1;
e s(e)
⊆ ∈ G
4. D µ−=a.e. R if 0 < s−1(v) < ; and
v e∈s−1(v) e | | ∞
S
5. for each e 1, there exist two measurable maps f : D R and
e r(e) e
∈ G →
f−1 : R D such that f f−1 µ−=a.e. id ,f−1 f µ−=a.e. id , the
e e → r(e) e◦ e Re e ◦ e Dr(e)
pushforward measure µ f , of f−1 in D , is absolutely continuous
◦ e e r(e)
with respect to µ in D , and the pushforward measure µ f−1, of
r(e) ◦ e
f in R , is absolutely continuous with respect to µ in R . Denote the
e e e
Radon-Nikodym derivative d(µ f )/dµ by Φ and the Radon-Nikodym
◦ e fe
derivative d(µ f−1)/dµ by Φ .
◦ e fe−1
We call R ,D ,f a -branching system on (X,µ).
e A e e∈G1,A∈G0
{ } G
It follows that µ in D is absolutely continuous with respect to µ f ,µ
r(e) e
◦
in R is absolutely continuous with respect to µ f−1,Φ > 0 µ-a.e. in
e ◦ e fe
D ,Φ > 0 µ-a.e. in R , and Φ (x)Φ (f (x)) = 1 µ a.e. in D . For
r(e) fe−1 e fe fe−1 e − r(e)
α ∗ 0, define f := f f , and define f−1 := f−1 f−1.
∈ G \ G α α1 ◦ ···◦ αn α αn ◦ ···◦ α1
It is straightforward to see that µ f in D is absolutely continuous
◦ α r(αn)
with respect to µ in D , and µ f−1 in R is absolutely continuous with
r(αn) ◦ α α1
respect to µ in R . Denote the Radon-Nikodym derivative d(µ f )/dµ by
α1 ◦ α
Φ , and denote the Radon-Nikodym derivative d(µ f−1)/dµ by Φ .
fα ◦ α fα−1
Theorem 3.2 Let be an ultragraph. Then there exists a -branching sys-
G G
tem.
8
Proof. Let X := R and let µ be the Lebesgue measure onall Borel sets of R.
Weenumeratetheset 1 = e andthesetofsinksG0 = v : s−1(v ) =
G { i}i≥1 sink { i i
. For each i 1, define R := [i 1,i], and define D := [ i,1 i].
∅}i≥1 ≥ ei − vi − −
For v G0 with s−1(v) = , define D := R . Define D = . For
v e∈s−1(v) e ∅
∈ 6 ∅ ∪ ∅
A = 0, define D := D .
A v∈A v
6 ∅ ∈ G ∪
It is easy to see that R ,D satisfies Condition (1)–(4) of Def-
e A e∈G1,A∈G0
{ }
inition 3.1.
Fix e 1. We prove the existence of f ,f−1,Φ ,Φ . Write R =
∈ G e e fe fe−1 e
[n,n + 1] for some n 0. Suppose that D = m [n ,n + 1], where
≥ r(e) i=1 i i
n < n . For each i, let F : [n ,n +1] [n+(i 1)/m,n+i/m] be an
i i+1 i i i → −S
arbitrary increasing bijection in C1([n ,n +1]). Piecing together F ’s yields
i i i
f , and piecing together F−1’s yields f−1. The existence of Φ ,Φ follows
e i e fe fe−1
easily.
Suppose that D = ∞ [n ,n + 1], where n < n . For each i, let
r(e) i=1 i i i i+1
F : [n ,n +1] [n+1 (1/2)i−1,n+1 (1/2)i] be an arbitrary increasing
i i i → −S −
bijection in C1([n ,n +1]). Piecing together F ’s yields f . Piecing together
i i i e
F−1’s and giving an arbitrary value at n + 1 yield f−1. The existence of
i e
Φ ,Φ follows easily. So we are done. (cid:3)
fe fe−1
In[25]itisshownthatthereexistsultragraphC*-algebrasthatareneither
Exel-Laca nor graph C*-algebras. In particular, the following example is
considered:
Example 3.3 Let be the ultragraph where 1 = ei,gi i∈N, G0 = w
G G { } { } ∪
vi i∈N and with the following range and source maps: r(gi) = G0 w for
{ } \{ }
each i N, r(e ) = v ,v ,v ,v ,... for each 1 i 3, r(e ) = v ,v
i i 4 5 6 i i i−3
∈ { } ≤ ≤ { }
for each i 4, s(g ) = w for each i N and s(e ) = v for each i N.
i i i
≥ ∈ ∈
Since the ultragraph C*-algebra associated to the ultragraph above is
not an Exel-Laca nor a graph C*-algebra, it is interesting to construct a
branching system associated to . We will define a concrete branching
G G−
system in R, with Lebesgue measure.
Define, for each i N, R = [i 1,i), R = [ i, i+1), D = [i 1,i)
∈ ei − gi − − vi −
and D = ( ,0). Now, defining D = D , for each A 0, we obtain
w A u
−∞ ∈ G
u∈A
S
9
that: D = [i 1,i) [3, )foreach1 i 3, D = [i 4,i 3) [i 1,i)
r(ei) − ∪ ∞ ≤ ≤ r(ei) − − ∪ −
for each i 4, and D = [0, ) for each i N. In the next figure we show
≥ r(gi) ∞ ∈
graphically an example of maps f−1 and f−1.
ei gi
❜
✻ f−1✁
e6✁
❜ ❜✁
f−1✁
e5✁
❜ ❜✁
f−1 f−1 f−1✁
e1 e2 e4✁
❜ ❜ ❜✁ r
✁
f−1 ✁f−1
r e3 r ✁ e6
✁ ✁
✁f−1 ✁f−1
r ✁ e2 r ✁ e5
f−1 f−1 f−1 ✁ ✁
g3 g2 g1 ✁f−1 ✁f−1
✁ e1 ✁ e4 ✲
R R R R R R R R R
g3 g2 g1 e1 e2 e3 e4 e5 e6
Remark 3.4 Notice that in the branching system above we have enumerated
the edges of differently from what we did in Theorem 3.2, but we kept the
G
main idea of how to define the measurable sets R and D .
e A
Remark 3.5 We will see, Theorem 4.1, that this branching system induces
a representation of C∗( ) in B( 2(R)) and, since satisfies condition (L),
G L G
this representation is faithful (see Theorem [24, Theorem 6.7]).
4 Representations of Ultragraph C*-algebras
on 2(X,µ) via Branching Systems
L
Next we show how to obtain a representation of an ultragraph C*-algebra
from a given Branching System.
10