$C^{\ast}$-Algebras associated with Mauldin-Williams Graphs

0.22 MB·English

by  Marius Ionescu

C -ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS
∗
GRAPHS
5
0
0
2 MARIUSIONESCUANDANDYASUOWATATANI

n
a Abstract. A Mauldin-Williams graph M is a generalization of an iterated
J functionsystem byadirectedgraph. Itsinvariantset K plays theroleofthe
9 self-similarset. We associate a C∗-algebraOM(K)withaMauldin-Williams
1 graph M and the invariant set K laying emphasis on the singular points.
We assume that the underlying graph G has no sinks and no sources. If M
] satisfies the open set condition in K and G is irreducible and is not a cyclic
A permutation, then the associated C∗-algebra OM(K) is simple and purely
O infinite. Wecalculate theK-groupsforsomeexamples includingthe inflation
ruleofthePenrosetilings.
.
h
t
a
m
1. Introduction
[

Self-similar sets are often constructed as the invariant set of iterated function
2
v systems[2],[13],[19]. Manyotherexamples,suchastheinflationruleofthePenrose
0 tilings,showthatagraphdirectedgeneralizationofiteratedfunctionsystemsisalso
8 interestingandhasbeendevelopedasMauldin-Williamsgraphs[3],[11],[23]. Since
4
we can regard graph directed iterated function systems as dynamical systems, we
0
expect that there exist fruitful connections between Mauldin-Williams graphs and
1
4 C∗-algebras.
0 In [14] the first-named author defined a C -correspondence for a Mauldin-
∗
/ Williams graph and showedthat the Cuntz-Pimsner algebra X (see [12], [29])
h
M OX
t isisomorphictothe Cuntz-Kriegeralgebra G ([8])associatedwiththe underlying
a O
graphG. Inparticular,iftheMauldin-WilliamsgraphhasonevertexandN edges,
m
the Cuntz-Pimsner algebra is isomorphic to the Cuntz algebra ([7]), which
N
: OX O
v recoversaresultin[30]. The constructionis usefulbecause itgivesmanyexamples
i ofdifferentnon-self-adjointalgebraswhicharenotcompletelyisometricallyisomor-
X
phic, but have the same C -envelope ([25]) as shown by the first-named author in
∗
r
a [15].
On the other hand, Kajiwara and the second-named author introduced C -
∗
algebras associated with rational functions in [17] including the singular points
(i.e. branched points). They developed the idea to associate C -algebras with
∗
self-similar sets considering the singular points in [16]. In this paper we associate
another C -algebra (K) with a Mauldin-Williams graph and its invariant
∗
OM M
set K putting emphasis on the singular points as above. We show that the associ-
atedC -algebra (K)isnotisomorphictotheCuntz-Kriegeralgebra forthe
∗ G
OM O
underlying graph G in general. This comes from the fact that the singular points
cause the failure of the injectivity of the coding by the Markov shift for G. We
assume that the underlying graph G has no sinks and no sources. We show that
the associated C -algebra (K) is simple and purely infinite if satisfies the
∗
OM M
opensetconditioninK andifGis irreducibleandisnotacyclicpermutation. We
1
C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 2
calculatetheK-groupsforsomeexamplesincludingtheinflationruleofthePenrose
tilings. The C -algebras associated with tilings were first studied by Connes in [6]
∗
and also discussed by Mingo [24] and Anderson-Putnam [1]. But we do not know
the exact relatition between their constructions and ours.
Ourconstructionhasacommonflavorwithseveraltopologicalgeneralizationsof
graphC -algebras[21]byBrenken[5],Katsura[18],MuhlyandSolel[26]andMuhly
∗
and Tomforde [27]. Since graph directed iterated function systems are sometimes
obtained as continuous cross sections of expanding maps, C -algebras associated
∗
withintervalmapsintroducedbyDeaconuandShultzin[9]arecloselyrelatedwith
our construction. In a different point of view, Bratteli and Jorgensen studied a
relationbetweeniterated function systems andrepresentationofCuntz algebrasin
[4].
TheauthorswouldliketothankP.Muhly forsuggestingtheircollaborationand
giving them the chance to discuss this work at a conference at the University of
Iowa.
2. Mauldin-Williams graphs and the associated C -correspondence
∗
ByaMauldin-Williams graph wemeanasystem =(G, T ,ρ , φ ),
v v v E0 e e E1
where G = (E0,E1,r,s) is a graph with a finite sMet of vert{ices E0}, a∈ fini{te s}et∈of
edgesE1,arange maprandasource maps,andwhere T ,ρ and φ
v v v E0 e e E1
{ } ∈ { } ∈
are families such that:
(1) Each T is a compact metric space with a prescribed metric ρ , v E0.
v v
∈
(2) For e E, φ is a continuous map from T to T such that
e r(e) s(e)
∈
cρ (x,y) ρ (φ (x),φ (y)) cρ (x,y)
′ r(e) s(e) e e r(e)
≤ ≤
for some constants c,c satisfying 0<c <c<1 (independent of e) and all
′ ′
x,y T .
r(e)
∈
We shall assume, too, that the source map s and the range map r are surjective.
Thus, we assume that there are no sinks and no sources in the graph G.
In the particular case when we have only one vertex and N edges, we obtain a
so called iterated function system (K, φ ).
i i=1,...,N
{ }
Aninvariantlist associatedwithaMauldin-Williamsgraph =(G, T ,ρ , φ )
v v v E0 e e E1
is a family (K ) of compact sets such that K T for Mall v E0{, and }su∈ch { } ∈
v v E0 v v
that ∈ ⊂ ∈
K = φ (K ).
v e r(e)
e[∈E1
s(e)=v
Since eachφ is a contraction, hasa unique invariantlist(see [23, Theorem1]).
e
M
We set T := T and K := K and we call K the invariant set of the
v E0 v v E0 v
Mauldin-William∈s graph. ∈
S S
Intheparticularcasewhenwehaveonevertexvandnedges,i.e. inthesettingof
aniteratedfunction system,theinvariantsetistheuniquecompactsubsetK :=K
v
of T =T such that
v
K =φ (K) φ (K).
1 n
∪···∪
In this paper we forget about the ambient space T. That is, we consider that the
Mauldin-Williamsgraphis =(G, K ,ρ , φ ),withK = K
being the invariant set. M { v v}v∈E0 { e}e∈E1 v∈E0 v
S
C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 3
We say that a graph G=(E0,E1,r,s) is irreducible (or totally connected) if for
every v ,v E0 there is a finite path w such that s(w) = v and r(w) = v . We
1 2 1 2
∈
will assume in this paper that the graph G=(E0,E1,r,s) is irreducible and not a
cyclicpermutation. Thatis thereexists avertexv E0 andthere existtwoedges
∗
∈
e =e such that s(e )=s(e )=v .
1 2 1 2 ∗
6
Foranaturalnumberm,wedefineEm := w =(w ,...,w ):w E1 and r(w )=
1 m i i
{ ∈
s(w ) for all i = 1,...,m 1 . An element w Em is called a path of length
i+1
− } ∈
m. We extend s and r to Em by s(w)=s(w ) and r(w) =r(w ) for all w Em.
1 m
∈
We set E = Em and denote the length of a path w by l(w). We define also
∗ m 1
Em(v):= w E≥m : s(w)=v to be the set of paths of length m starting at the
{ S∈ }
vertex v and E (v)= Em(v) to be the set of finite paths starting at v.
∗ m 1
Theinfinitepathspace≥isE = w=(w ) : r(w )=s(w ) for all n 1
∞ n n 1 n n+1
S { ≥ ≥ }
and the space of infinite paths starting at a vertex v is E (v) = w E :
∞ ∞
{ ∈
s(w)=s(w )=v . OnE (v)wedefinethe metricδ (α,β)=cl(α β) ifα=β and
1 ∞ v ∧
} 6
0 otherwise, where α β is the longest common prefix of α and β (see [10, Page
∧
116]). Then E (v) is a compact metric space, and, since E equals the disjoint
∞ ∞
union of the spaces E (v), E is a compact metric space in a natural way. For
∞ ∞
w Em, let φ = φ φ and K = φ (K ). Then for any infinite path
∈ w w1 ◦··· wm w w r(w)
mαa=p(παn:)En∈N, Kn≥b1yK(απ1(,α...),αn=) containKs only one. pSoinincte.πT(Ehere)foisrealwsoe acannindveafirniaenat
∞ →T { } n 1 (α1,...,αn) ∞
set, we have that π(E ) = K. Th≥us π is a continuous, onto map. Moreover, for
∞
anyy K andanyneighborhoTodU K ofy,thereexistn Nandw En(v )
∈ v0 ⊂ v0 ∈ ∈ 0
such that
y φ (K ) U.
w r(w)
∈ ⊂
WesaythataMauldin-Williamsgraph satisfiestheopen setcondition in K if
M
thereexists afamily ofnon-emptysets(V ) , suchthatV K forallv E0,
v v E0 v v
and such that ∈ ⊂ ∈
φ (V ) V for all v E0,
e r(e) v
⊂ ∈
e[∈E1
s(e)=v
and φ (V ) φ (V )= if e=f.
e r(e) f r(f)
∅ 6
Then V := v E0Vv is an open d\ense subset of K. Moreover, for n ∈ N and
w,v En, if w∈=v and r(w)=r(v) then φ (V ) φ (V )= .
w r(w) v r(v)
∈ S 6 ∅
For e E1, we define the cograph of φ to be the set
e
∈ T
cograph(φ )= (x,y) K K : x=φ (y) K K.
e s(e) r(e) e
{ ∈ × }⊂ ×
We shall consider the union
= ( φ : e E1 ):= cograph(φ ).
e e
G G { ∈ }
e E1
∈[
ConsidertheC -algebraA=C(K)andletX =C( ). ThenXisaC -correspondence
∗ ∗
G
over A with the structure defined by the formulae:
(a ξ b)(x,y) = a(x)ξ(x,y)b(y),
· ·
ξ,η (y) = ξ(φ (y),y)η(φ (y),y),
A e e
h i
eX∈E1
y∈Kr(e)
for all a,b A, ξ,η X, (x,y) and y K. It is clear that the A-valued inner
∈ ∈ ∈G ∈
product is well defined. The left multiplication is given by the -homomorphism
∗
C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 4
Φ : A (X) such that (Φ(a)ξ)(x,y) = a(x)ξ(x,y) for a A and ξ X. Put
→ L ∈ ∈
ξ = ξ,ξ 1/2.
2 A
k kFor aknhy niatku∞ral number n, we define = ( φ : w En ) and a C -
n w ∗
G G { ∈ }
correspondence X =C( ) similarly. We also define a path space of length n
n n n
G P
by
= (φ (y),φ (y),...,φ (y),y) Kn+1 :
Pn { w1,···,wn w2,···,wn wn ∈
w =(w ,...,w ) En,y K .
1 n r(w)
∈ ∈ }
Then Y :=C( ) is a C -correspondence over A with an A-valued inner product
n n ∗
P
defined by
ξ,η (y)= ξ(φ (y),...,φ (y),y)η(φ (y),...,φ (y),y),
h iA w1,...,wn wn w1,...,wn wn
w∈En
y∈XKr(w)
for all ξ,η Y and y K.
n
∈ ∈
Proposition 2.1. Let = (G, K ,ρ , φ ) be a Mauldin-Williams
v v v E0 e e E1
graph. Let K be the invMariant set{. Then}X∈ ={C(})∈is a full C -correspondence
∗
G
over A = C(K) without completion. The left action Φ : A (X) is unital and
→ L
faithful. Similar statements hold for Y =C( ).
n n
P
Proof. For any ξ X we have
∈
1/2
ξ ξ 2 =sup ξ(φe(y),y)2 √N ξ ,
k k∞ ≤k k y K | | ≤ k k∞
 ∈ y∈eX∈KEr(1e) 
 
where N is the number of edges in E1. Therefore the norms and are
2
k k k k∞
equivalent. Since C( ) is complete with respect to , it is also complete with
G k k∞
respect to .
2
kk
Let ξ X be defined by the formula
∈
1
ξ(x,y)= for all (x,y) .
#(e:y K ) ∈G
r(e)
∈
Then ξ,ξ (y)=1,hence pX,X containsthe identity ofA. ThereforeX is full.
A A
h i h i
If a A is not zero, then there exists x K such that a(x )=0. Since K is the
0 0
∈ ∈ 6
invariant set of the Mauldin-Williams graph, there exists e E1 and y K
0 r(e)
∈ ∈
suchthatx K andφ (y )=x . Chooseξ X suchthatξ(x ,y )=0. Then
0 s(e) e 0 0 0 0
Φ(a)ξ =0, he∈nce Φ is faithful. The statements f∈or Y are similarly prov6ed. (cid:3)
n
6
Definition 2.2. Let = (G, K ,ρ , φ ) be a Mauldin-Williams
v v v E0 e e E1
graph with the invariaMnt set K.{ We a}ss∈ociat{e a}C∈ -algebra (K) to as
∗
OM M
the Cuntz-Pimsner algebra of the C -correspondence X = C( ) over the C -
X ∗ ∗
O G
algebra A=C(K).
As in [16], we denote by alg the -algebrageneratedalgebraicallyby A andS
with ξ X. The gauge actiOonXis γ :∗R Aut defined by γ (S )=eitS for alξl
X t ξ ξ
∈ → O
ξ X, and γ (a)=a for all a A.
t
∈ ∈
C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 5
Proposition 2.3. Let = (G, K ,ρ , φ ) be a Mauldin-Williams
v v v E0 e e E1
graph. Assume that K iMs the inva{riant se}t o∈f the{gr}ap∈h. Then there is an isomor-
phism ϕ :X n C( ) as C -correspondences over A such that
n ⊗ n ∗
→ P
(ϕ (ξ ξ ))(φ (y),...,φ (y),y)
n 1⊗···⊗ n w1,...,wn wn
=ξ (φ (y),φ (y))ξ (φ (y),φ (y))...ξ (φ (y),y)
1 w1,...,wn w2,...,wn 2 w2,...,wn w3,...,wn n wn
for all ξ ,...,ξ X, y K, and w = (w ,...,w ) En such that y K .
1 n ∈ ∈ 1 n ∈ ∈ r(wn)
Moreover, let ρ : be an onto continuous map such that
n n n
P →G
ρ (φ (y),...,φ (y),y)=(φ (y),y).
n w1,...,wn wn w1,...,wn
Then ρ : C( ) f f ρ C( ) is an embedding as a Hilbert submodule
∗n Gn ∋ → ◦ n ∈ Pn
preserving inner product.
Proof. Itiseasyto seethatϕ is well-definedandabimodule morphism. Weshow
n
that ϕ preserves inner product. Consider the case when n = 2 for simplicity of
n
the notation. Let ξ ,ξ ,η ,η X. We have
1 2 1 2
∈
ξ ξ ,η η (y) = ξ , ξ ,η η (y)
1 2 1 2 A 2 1 1 A 2 A
h ⊗ ⊗ i h h i · i
= ξ (φ (y),y) ξ ,η (φ (y))η (φ (y),y)
2 e 1 1 A e 2 e
h i
e∈E
y∈XKr(e)
= ξ (φ (y),φ (y))ξ (φ (y),y)η (φ (y),φ (y))η (φ (y),y)
1 fe e 2 e 1 fe e 2 e
yf∈XeK∈rE(e2)
= ϕ (ξ ξ ),ϕ (η η ) (y).
2 1 2 2 1 2 A
h ⊗ ⊗ i
Since ϕ preservesthe inner product, it is one to one. Using the Stone-Weierstrass
n
Theorem, one can show that ϕ is also onto. The rest is clear. (cid:3)
n
We let i : C( ) C( ) be the natural inner-product preserving embed-
n,m n m
P → P
ding, for m n.
≥
Definition 2.4. Consider a covering map π : K defined by π(x,y) = y for
G →
(x,y) . Define the set
∈G
B( ):= x K : x=φ (y)=φ (y) for some y K and e=f .
e f
M { ∈ ∈ 6 }
The set B( ) will be described by the ideal I :=Φ 1( (X)) of A. We define a
X −
M K
branch index e(x,y) at (x,y) by
∈G
e(x,y):=# e E1 : φ (y)=x .
e
{ ∈ }
Hence x B( ) if and only if there exists some y K with e(x,y) 2. For
∈ M ∈ ≥
x K we define
∈
I(x):= e E1 : there exists y K such that x=φ (y) .
e
{ ∈ ∈ }
Lemma 2.5. In the above situation, if x K B( ), then there exists an open
∈ \ M
neighborhood U of x satisfying the following:
x
(1) U B( )= .
x
M ∅
(2) If e I(x), then φ (φ 1(U )) U = for e=f, such that r(e)=r(f).
T∈ f −e x x ∅ 6
(3) If e / I(x), then U φ (K )= .
x e r(e)
∈ T ∅
T
C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 6
Proof. Let x K B( ). Let v E0 such that x K . Since B( ) and
∈ \ M 0 ∈ ∈ v0 M
φ (K ) are closed and x is not in either of them, there exists an open
e/I(x) e r(e)
nei∈ghbourhood W K of x such that
S x ⊂ v0
W (B( ) φ (K ))= .
x e r(e)
M ∪ ∅
\ e∈/[I(x)
For each e I(x) there exists a unique y K with x = φ (y ), since x / B( ).
e e e
∈ ∈ ∈ M
For f E1, if r(e) = r(f) and f = e then φ (y ) = φ (y ) = x. Therefore there
f e e e
∈ 6 6
exists anopen neighborhoodVe ofx such that φ (φ 1(Ve)) Ve = if f =e and
x f −e x x ∅ 6
r(f)=r(e). Let U :=W ( Ve). Then U is an open neighborhood of x
and satisfies all thexrequirxementse∈.I(x) x x T (cid:3)
T T
Proposition 2.6. Let = (G, K ,ρ , φ ) be a Mauldin-Williams
v v v E0 e e E1
graph. Assume that theMsystem {satisfie}s∈the o{pen}s∈et condition in K. Then
M
I = a A=C(K) : a vanishes on B( ) .
X
{ ∈ M }
Proof. The proof requiresonly minor modifications fromthe proof of [16, Proposi-
tion 2.4]. (cid:3)
Corollary 2.7. #B( )=dim(A/I ).
X
M
Corollary 2.8. The closed set B( ) is empty if and only if Φ(A) is contained in
M
(X) if and only if X is finitely generated projective right A module.
K
3. Simplicity and Pure Infinitness
Let = (G, K ,ρ , φ ) be a Mauldin-Williams graph. Let A =
v v v E0 e e E1
C(K) aMnd X =C{( ). Fo}r∈e E{1 d}e∈fine an endomorphism β :A A by
e
G ∈ →
a(φ (y)) if y K
(β (a))(y):= e ∈ r(e)
e
(0 otherwise,
for all a A and y K. We also define a unital completely positive map E :
∈ ∈ M
A A by
→
1
(E (a))(y): = a(φ (y))
M # e E1 :y Kr(e) e
{ ∈ ∈ } eX∈E1
y∈Kr(e)
1
= β (a)(y),
# e E1 :y K e
r(e)
{ ∈ ∈ } eX∈E1
y∈Kr(e)
for a A, y K. For the function ξ X defined by the formula
0
∈ ∈ ∈
1
ξ (x,y)=
0
# e E1 :y K
r(e)
{ ∈ ∈ }
we have
p
E (a)= ξ ,Φ(a)ξ andE (I)= ξ ,ξ =I.
0 0 A 0 0 A
M h i M h i
Let D :=S (K).
ξ0 ∈OM
Lemma 3.1. In the above situation, for a A, we have that D aD =E (a) and
∗
in particular D D =I. ∈ M
∗
Proof. The same with [16, Lemma 3.1]. (cid:3)
C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 7
Definition 3.2. Let = (G, K ,ρ , φ ) be a Mauldin-Williams
v v v E0 e e E1
graph. We say that anMelement a{ A=C}(∈K) i{s ( } ,∈En)-invariant if
∈ M
a(φ (y))=a(φ (y))for anyy K andα,β En such thaty K =K .
α β r(α) r(β)
∈ ∈ ∈
If a A is ( ,En)-invariant then a is also ( ,En 1)-invariant. Then, similar
−
∈ M M
like in [16, Definition on page 11], if a is ( ,En)-invariant, we can define
M
βk(a)(y):=a(φ ...φ (y)), for anyw Ek, such thaty K .
w1 wn ∈ ∈ r(wn)
Then, for any ξ ,...,ξ X and a A ( ,En)-invariant , we have the relation:
1 n
∈ ∈ M
aS ...S =S ...S βn(a).
ξ1 ξn ξ1 ξn
Lemma 3.3. Let =(G, K ,ρ , φ ) be a Mauldin-Williams graph
v v v E0 e e E1
such that (E0,E1,Mr,s) is an{irreduc}ib∈le gr{aph}. ∈For any non-zero positive element
a A and for every ε > 0 there exists n N and ξ X n with ξ,ξ = 1 such
⊗ A
∈ ∈ ∈ h i
that
a ε S aS a .
k k− ≤ ξ∗ ξ ≤k k
Proof. Letx K besuchthat a =a(x ). Letv E0suchthatx K . Then
0 ∈ k k 0 0 ∈ 0 ∈ v0
thereexistsanopenneighborhoodU ofx inK suchthatforanyx U wehave
0 0 v0 ∈ 0
a ε a(x) a . Let U anopen neighborhoodof x in K and K compact
k k− ≤ ≤k k 1 0 v0 1
suchthatU K U . Sincethemapπ :E K isontoandcontinuous,there
1 1 0 ∞
exists some n⊂1 N⊂and α En1(v0) such that→φα(Kr(α)) U1. For any vertex
∈ ∈ ⊂
v V, since the graph G is irreducible, there exists a path w E from r(α) to
v ∗
∈ ∈
v. Then φ (K ) U . Hence, for each v V, there is α En(v)(v ), for some
αwv v ⊂ 1 ∈ v ∈ 0
n(v) n , suchthat φ (K ) U . For each v V, define the closed subsets F
≥ 1 αv v ⊂ 1 ∈ 1,v
and F of K K by
2,v
×
F = (x,y) K K : x=φ (y),x K ,y K ,α En(v)(v )
1,v α 1 v 0
{ ∈ × ∈ ∈ ∈ }
F = (x,y) K K : x=φ (y),x Uc,y K ,α En(v)(v ) .
2,v { ∈ × α ∈ 0 ∈ v ∈ 0 }
Since F F = for all w V and F F = if v = w, there exists
1,v 2,w 1,v 1,w
∩ ∅ ∈ ∩ ∅ 6
g C( ) such that 0 g (x,y) 1 and
v n(v) v
∈ G ≤ ≤
1 if (x,y) F
1,v
g (x,y)= ∈
v
(0 if (x,y) w E0F2,w w=vF1,w.
∈∪ ∈ ∪ 6
Since φαv(Kv)⊂U1 for eachy ∈Kv, there exists xy ∈SU1 suchthatxy =φαv(y)∈
U K . Therefore
1 1
⊂
(3.1) g ,g (y)= g (φ (y),y)2 g (x ,y)2 =1
v v A v α v y
h i | | ≥| |
α∈XEn(v)
y∈Kr(α)
for all y K . Let n =max n(v) : v E0 . We identify X n with C( ) as in
v ⊗ n
∈ { ∈ } P
Proposition 2.3. We denote i (ρ (g )) C( ) also by g for each v V.
n(v),n ∗n(v) v ∈ Pn v ∈
Let g := g C( ). Then g,g (y) 1 for all y K. Thus b := g,g
v V v ∈ Pn h iA ≥ ∈ h iA
is positive an∈d invertible. Let ξ :=gb 1/2 X n. Then
P − ∈ ⊗
ξ,ξ = gb 1/2,gb 1/2 =b 1/2 g,g b 1/2 =1 .
A − − A − A − A
h i h i h i
For anyy K and any α En suchthat y K , let x=φ (y). If x U , then
r(α) α 0
∈ ∈ ∈ ∈
a ε a(x), and, if x Uc, then
k k− ≤ ∈ 0
ξ(φα1,...,αn(y),...,φαn(y),y)=g(x,y)b−1/2(y)=0.
C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 8
Therefore
a ε = ( a ε) ξ,ξ (y)
A
k k− k k− h i
= ( a ε) ξ(φ (y),...,φ (y),y)2
k k− | α1,...,αn αn |
α∈En
y∈XKr(α)
a(φ (y))ξ(φ (y),...,φ (y),y)2
≤ α | α1,...,αn αn |
α∈En
y∈XKr(α)
= hξ,aξiA(y)=Sξ∗aSξ(y).
We also have that
Sξ∗aSξ =hξ,aξiA ≤kakhξ,ξiA =kak.
(cid:3)
Lemma 3.4. Let =(G, K ,ρ , φ ) be a Mauldin-Williams graph.
v v v E0 e e E1
Assume that K is tMhe invari{ant set }of∈the g{rap}h.∈For any non-zero positive element
a A and for any ε > 0 with 0 < ε < a , there exists n N and u X n such
⊗
∈ k k ∈ ∈
that
kuk2 ≤(kak−ε)−1/2 andSu∗aSu =1.
Proof. The proof is identical with the proof of [16, Lemma 3.4]. (cid:3)
Lemma 3.5. Let =(G, K ,ρ , φ ) be a Mauldin-Williams graph.
v v v E0 e e E1
Suppose that the grMaph G ha{s no sin}ks∈and{no}so∈urces and it is irreducible and not
a cyclic permutation. Assume that K is the invariant set of the graph and that
satisfies the open set condition in K. For any n N, for any T (X n), and fMor
⊗
∈ ∈L
any ε>0 there exists a positive element a A such that a is ( ,En)-invariant,
∈ M
Φ(a)T 2 T 2 ε,
k k ≥k k −
and βp(a)βq(a)=0 for p,q =1,...,n with p=q.
6
Proof. Let n N, let T (X n), and let ε>0. Then there exists ξ X n such
⊗ ⊗
∈ ∈L ∈
that ξ =1 and T 2 Tξ 2> T 2 ε. Hence thereexistsy K forsome
k k2 k k ≥k k2 k k − 0 ∈ v0
v V such that
0
∈
Tξ 2= (Tξ)(φ (y ),...,φ (y ),y )2 > T 2 ε.
k k2 | α1,...,αn 0 αn 0 0 | k k −
α∈En
r(Xα)=v0
Then there exists an open neighborhood U of y in K such that for any y U
0 0 v0 ∈ 0
(Tξ)(φ (y),...,φ (y),y)2 > T 2 ε.
| α1,...,αn αn | k k −
α∈En
r(Xα)=v0
Since satisfies the open set condition in K, there exists a family of non-empty
M
sets (V ) , such that V K , for all v E0,
v v E0 v v
∈ ⊂ ∈
φ (V ) V for all v E0,
e r(e) v
⊂ ∈
e[∈E1
s(e)=v
and φ (V ) φ (V )= if e=f.
e r(e) f r(f)
∅ 6
MThoerneotvheerr,etehxeirsetsisy1so∈mVevk0∩′ U0Nanadnda\n(eo1p,e.n..n,eeikg′h)borEhko′o(vd0U)1suocfhy1thwaitthU1 ⊂V∩U0.
∈ ∈
φ (V ) U V U .
e1,...,ek′ r(ek′) ⊂ 1 ⊂ v0 ∩ 0
C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 9
Since the graph G is not a cyclic permutation, there is a vertex v E0 and two
∗
∈
edges e,e E1 such that e = e . Since the graph G is irreducible, there exists
′ ′′ ′ ′′
∈ 6
a path from r(ek′) to v∗. Hence we have a path (e1,...,ek) Ek(v0) for some
k N, k k , such that r(e )=v and φ (V ) U ∈V U . Then we
∈ ≥ ′ k ∗ e1,...,ek r(ek) ⊂ 1 ⊂ v0∩ 0
can find a path (e ,...,e ) En(v ) such that e = e if i = 1. To see
k+1 k+n ∗ k+1 k+i
∈ 6 6
this, let e = e. If r(e ) = v , take e = e . If r(e ) = v , since G has
k+1 ′ k+1 ∗ k+2 ′′ k+1 ∗
6
no sinks, there is an edge e E1 such that s(e) = r(e ). Then e = e . Let
k+1 k+1
∈ 6
e = e. If r(e ) = v , take e = e ; if r(e ) = v , take e to be any
k+2 k+2 ∗ k+3 ′′ k+2 ∗ k+3
6
edge such that s(e ) = r(e ). Therefore e = e . Inductively, we obtain
k+3 k+2 k+3 k+1
6
the path (e ,...,e ) En(v ) with the desired property. Then
k+1 k+n ∗
∈
=φ (V ) U V U .
∅6 e1,...,ek+n r(ek+n) ⊂ 1 ⊂ v0 ∩ 0
There exist y U , an open neighborhood U of y in K and a compact set L
2 ∈ 1 2 2 v0
such that
y U L φ (V ) U V U .
2 ∈ 2 ⊂ ⊂ e1,...,ek+n r(ek+n) ⊂ 1 ⊂ v0 ∩ 0
Let b A such that 0 b 1, b(y2) = 1 and bUc = 0. For α En such that
∈ ≤ ≤ | 2 ∈
r(α)=v , we have
0
φ (y ) φ (U ) φ (L) φ (V ).
α 2 ∈ α 2 ⊆ α ⊆ α v0
Moreover,for α,β En such that r(α)=r(β)=v , by the open set condition,
0
∈
φ (L) φ (L)= ifα=β.
α β
∩ ∅ 6
We define a positive function a on K by the formula
b(φ 1(x)) ifx φ (L), α En such thatr(α)=v
a(x)= −α ∈ α ∈ 0
(0 otherwise.
ThenaiscontinuousonK,hencea A. Byconstruction,ais( ,En)-invariant.
∈ M
Let p n be a natural number. Let (α ,...,α ) Ep. If there is no path
1 p
≤ ∈
(α ,...,α ) En p(r(α )) such that r(α ) = v , then β ...β (a) = 0. If
p+1 n ∈ − p n 0 αp α1
thereis atleastone path(α ,...,α ) En p(r(α ))suchthatr(α )=v ,then
p+1 n − p n 0
∈
supp(β ...β (a)) φ (supp(b)).
αp α1 ⊆ αp+1...αn
(αp+1,...,αn[)∈En−p(r(αp))
r(αn)=v0
Since supp(b) L φ (V ) we have that
⊆ ⊂ e1...ek+n r(ek+n)
supp(β ...β (a)) φ φ (V ).
αp α1 ⊆ αp+1...αn e1...ek+n r(ek+n)
(αp+1,...,αn[)∈En−p(r(αp))
r(αn)=v0
Then, for 1 p<q n we have that
≤ ≤
supp(βp(a)) φ φ (V )
⊆ αp+1...αn e1...ek+n r(ek+n)
(αp+1,...[,αn)∈En−p
r(αn)=v0
and
supp(βq(a)) φ φ (V ).
⊆ αq+1...αn e1...ek+n r(ek+n)
(αq+1,...[,αn)∈En−q
r(αn)=v0
Since (n p)+(k+1)-th subsuffixes are different as e = e , we have
k+1 k+1+(q p)
that supp−(βp(a)) supp(βq(a))= . Hence βp(a)βq(a)=0.6 −
∩ ∅
C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 10
Moreover,we have
Φ(a)Tξ 2 = sup (a(φ (y))(Tξ)(φ (y),...,φ (y),y) 2
k k2 | α α αn |
y∈K y∈αXK∈Er(nα)
= sup (b(y))(Tξ)(φ (y),...,φ (y),y) 2
| α αn |
y∈Ly∈αXK∈Er(nα)
(Tξ)(φ (y ),...,φ (y ),y ) 2
≥ | α 2 αn 2 2 |
α∈En
y2∈XKr(α)
> T 2 ε,
k k −
because y U . Thus Φ(a)T 2 T 2 ε. (cid:3)
2 0
∈ k k ≥k k −
As in [16], we let be the C -subalgebra of generated by (X k), k =
n ∗ X ⊗
F F K
0,1,...,n and B be the C -subalgebra of generated by
n ∗ X
O
n
{Sx1···SxkSy∗k...Sy∗1 : x1,...,xk,y1,...,yk ∈X}∪A.
i=1
[
We will also use the isomorphism ϕ: B such that
n n
F →
ϕ(θx1⊗···⊗xk,y1⊗···⊗yk)=Sx1...SxkSy∗k...Sy∗1.
Lemma 3.6. In the above situation, let b=c c for some c alg. We decompose
∗ ∈OX
b = b with γ (b ) = eijtb . For any ε > 0 there exists P A with 0 P I
j j t j j ∈ ≤ ≤
satisfying the following:
P
(1) Pb P =0 (j =0).
j
6
(2) Pb P b ε.
0 0
k k≥k k−
Proof. The proof requires only small modifications from the proof of [16, Lemma
3.6]. (cid:3)
Having proved the equivalent of the [16, Lemma 3.1-Lemma 3.6], we obtain,
usingthesameproofas[16,Theorem3.7],thecorrespondingresultfortheMauldin-
Williams graph:
Theorem3.7. Let =(G, K ,ρ , φ )beaMauldin-Williamsgraph.
v v v E0 e e E1
Suppose that the graMph G has{no sink}s∈and{no }so∈urces, it is irreducible, and is not
a cyclic permutation. Assume that K is the invariant set of the Mauldin-Williams
graph and that satisfies the open set condition in K. Then the associated C -
∗
M
algebra (K) is simple and purely infinite.
OM
Using the same argument as in [16, Proposition3.8] one can show that (K)
OM
is separableandnuclear,andsatisfiesthe UniversalCoefficientTheorem. Thus, by
the classificationtheorem of Kirchbergand Phillips [20, 28], the isomorphismclass
of (K) is completely determined by the K-theory with the class of the unit.
OM
4. Examples
We will compute the K-groups of the C -algebra associated with a graph G =
∗
(E0,E1,r,s) using the fact that K (C (G)) is isomorphic to the kernelof 1 At :
1 ∗ − G