Unique tracial state on the labeled graph C ∗ -algebra associated to Thue--Morse sequence
UUNIQUE TRACIAL STATE ON THE LABELED GRAPH C ∗ -ALGEBRA ASSOCIATED TO THE THUE–MORSE SEQUENCE SUN HO KIM
Abstract.
We give a concrete formula for the unique faithful trace on the finitesimple non-AF labeled graph C ∗ -algebra C ∗ ( E Z , L , E Z ) associated to the Thue–Morse sequence ( E Z , L ). Our result provides an alternative proof of the existenceof a labeled graph C ∗ -algebra that is not Morita equivalent to any graph C ∗ -algebras. Furthermore, we compute the K -groups of C ∗ ( E Z , L , E Z ) using thepath structure of the Thue–Morse sequence. Introduction
In order to study the relationship between the class of C ∗ -algebras and the classof dynamical systems, Cuntz and Krieger [6] introduced C ∗ -algebras associatedto shifts of finite type. Since then during the past thirty years there have beenmany generalizations of the Cuntz-Krieger algebras via a lot of different approaches.These generalizations include, for example, graph C ∗ -algebras C ∗ ( E ) of directedgraphs E (see [4, 8, 15, 16] among others), Exel-Laca algebras O A of infinite { , } -matrices A [7], ultragraph C ∗ -algebras C ∗ ( G ) [20], and Matsumoto algebras O Λ , O Λ ∗ of one-sided shift spaces Λ over finite alphabets [17]. To unify aforementionedalgebras, Bates and Pask [2] introduced a class of C ∗ -algebras C ∗ ( E, L , E ) associatedto labeled spaces ( E, L , E ), which we call the labeled graph C ∗ -algebras.It is now well known that simple graph C ∗ -algebras are classifiable by their K-theories (see [19, Theorem 3.2] and [18, Remark 4.3]), and are either AF or purelyinfinite (see [15, Corollary 3.10]). Bates and Pask showed in [3] that there existsa unital simple purely infinite labeled graph C ∗ -algebra whose K -group is notfinitely generated. Since the K -group of a unital graph C ∗ -algebra is always finitelygenerated, this example shows that the class of labeled graph C ∗ -algebras is strictlylarger than the class of graph C ∗ -algebras up to isomorphism. On the other hand,the three classes of graph C ∗ -algebras, Exel-Laca algebras, and ultragraph C ∗ -algebras are known to be equal up to Morita equivalence [13], and so the followingquestion naturally arises: “Up to Morita equivalence, is the class of labeled graph C ∗ -algebras still larger than the class of graph C ∗ -algebras?” A negative answer tothe question is given very recently in [12] by showing that there exist unital simplefinite labeled graph C ∗ -algebras which are neither AF nor purely infinite. (Notehere that the property of being AF or purely infinite, simple is preserved underMorita equivalence.) Mathematics Subject Classification.
Key words and phrases. labeled graph C ∗ -algebra, finite C ∗ -algebra, Thue–Morse sequence.Research supported by BK21 PLUS SNU Mathematical Sciences Division. a r X i v : . [ m a t h . OA ] M a r S. H. KIM
To show the existence of such a finite simple unital non-AF labeled graph C ∗ -algebra, it is proven in [12] that there exists a family of non-AF simple unitallabeled graph C ∗ -algebras with traces, more precisely, the algebras in the familyare obtained as crossed products of Cantor minimal subshifts. If a Cantor minimalsubshift is uniquely ergodic (for example, the subshift of Thue–Morse sequence isuniquely ergodic), its corresponding labeled graph C ∗ -algebra has a unique trace. Inthis paper, we provide another proof for the existence and uniqueness of a trace onthe labeled graph C ∗ -algebra of the Thue–Morse sequence, and we give a concreteformula of the trace. We also obtain K -groups of this labeled graph C ∗ -algebraassociated to the Thue–Morse sequence.The paper is organized as follows. In Section 2, we review the definitions of alabeled graph C ∗ -algebra and the Thue–Morse sequence ω . We also describe thestructure of the AF-core of the labeled graph C ∗ -algebra C ∗ ( E Z , L , E Z ) associatedto the Thue–Morse sequence. Then in Section 3, we discuss properties of labeledpaths of ( E Z , L ) by using a new notation which helps us to avoid complicated com-putations. In Section 4 we prove the existence of the unique trace on C ∗ ( E Z , L , E Z )and give a concrete formula (see Theorem 4.3 and Proposition 4.4). Finally, inSection 5 we give a computation of K -groups and in Section 6 a representation on (cid:96) ( Z ) of C ∗ ( E Z , L , E Z ). 2. Preliminaries
Labeled graph C ∗ -algebras. We follow the notational convention of [15] forgraphs and of [1, 3] for labeled graphs and their C ∗ -algebras.A (directed) graph E = ( E , E , r, s ) consists of a vertex set E , an edge set E ,range and source maps r, s : E → E . A path λ is a sequence of edges λ λ . . . λ n with r ( λ i ) = s ( λ i +1 ) for i = 1 , . . . , n −
1. The length of a path λ = λ λ . . . λ n is | λ | := n and we denote by E n the set of all paths of length n . The range and sourcemaps can be naturally extended to E n by r ( λ ) := r ( λ n ), s ( λ ) := s ( λ ). We alsodenote by E ∗ = ∪ n ≥ E n the set of all vertices and all finite paths in E . A vertex iscalled a source if it receives no edges and a sink if it emits no edges. Assumption . Throughout the paper, we assume that a graph has no sinks orsources.Let A be an alphabet set. A labeling map L is a map from E onto A . For apath λ = λ λ . . . λ n ∈ E ∗ \ E , we define it labeling L ( λ ) to be L ( λ ) . . . L ( λ n ) anddenote by L ∗ ( E ) the set ∪ n ≥ L ( E n ) of all labeled paths. Given a graph E and alabeling map L , we call ( E, L ) a labeled graph . The range of a labeled path α isdefined by r ( α ) := { r ( λ ) ∈ E : L ( λ ) = α } , and the relative range of α with respect to A ⊂ E is defined by r ( A, α ) := { r ( λ ) ∈ E : s ( λ ) ∈ A and L ( λ ) = α } . Let B be a subset of 2 E . If B contains all ranges of labeled paths, and isclosed under finite union, finite intersection, and relative range, then we call B an accommodating set for ( E, L ) and a triple ( E, L , B ) a labeled space . RACE ON THE THUE–MORSE LABELED GRAPH C ∗ -ALGEBRA 3 For
A, B ⊂ E and 1 ≤ n ≤ ∞ , let AE n = { λ ∈ E n : s ( λ ) ∈ A } , E n B = { λ ∈ E n : r ( λ ) ∈ B } . A labeled space ( E, L , B ) is said to be set-finite ( receiver set-finite , respectively) iffor every A ∈ B and 1 ≤ l < ∞ the set L ( AE l ) ( L ( E l A ), respectively) is finite.If a labeled space ( E, L , B ) satisfies r ( A, α ) ∩ r ( B, α ) = r ( A ∩ B, α )for all
A, B ∈ B , then ( E, L , B ) is called weakly left-resolving . We will only considerthe smallest accommodating set E which is non-degenerate (namely closed underrelative complement). We also assume that ( E, L , E ) always is set-finite, receiverset-finite and weakly left-resolving.We define an equivalence relation ∼ l on E ; we write v ∼ l w if L ( E ≤ l v ) = L ( E ≤ l w ) as in [3]. The equivalence class [ v ] l of v is called a generalized vertex . If k > l , [ v ] k ⊂ [ v ] l is obvious and [ v ] l = ∪ mi =1 [ v i ] l +1 for some vertices v , . . . , v m ∈ [ v ] l .Moreover every set A ∈ E is the finite union of generalized vertices, that is, A = ∪ ni =1 [ v i ] l (1)for some v i ∈ E , l ≥
1, and n ≥ Definition 2.2. ([1, Definition 2.1]) Let ( E, L , E ) be a labeled space. We define a representation of a labeled space ( E, L , E ) to be a set consisting of partial isometries { s a : a ∈ A} and projections { p A : A ∈ E} such that for a, b ∈ A and A, B ∈ E ,(i) p ∅ = 0, p A p B = p A ∩ B , and p A ∪ B = p A + p B − p A ∩ B ,(ii) p A s a = s a p r ( A,a ) ,(iii) s ∗ a s a = p r ( a ) and s ∗ a s b = 0 if a (cid:54) = b ,(iv) for each A ∈ E , p A = (cid:88) a ∈L ( AE ) s a p r ( A,a ) s ∗ a . It was proven in [2, Theorem 4.5] that there always exists a universal representationfor a labeled space. We call a C ∗ -algebra generated by a universal representationof ( E, L , E ) a labeled graph C ∗ -algebra C ∗ ( E, L , E ). Remark . Let ( E, L , E ) be a labeled space.(i) By [2, Lemma 4.4], it follows that C ∗ ( E, L , E ) = span { s α p A s ∗ β : α, β ∈ L ∗ ( E ) , A ∈ E} = span { s α p r ( µ ) s ∗ β : α, β, µ ∈ L ∗ ( E ) } . Moreover, from r ( µ ) = ∪ a ∈A r ( aµ ), we may assume that | µ | > | α | , | β | .(ii) Universal property of C ∗ ( E, L , E ) = C ∗ ( s a , p A ) defines a strongly continuousaction γ : T → Aut( C ∗ ( E, L , E )), called the gauge action , given by γ z ( s a ) = zs a and γ z ( p A ) = p A for a ∈ A and A ∈ E . S. H. KIM (iii) Let C ∗ ( E, L , E ) γ denote the fixed point algebra of the gauge action. It iswell known that C ∗ ( E, L , E ) γ is an AF algebra and C ∗ ( E, L , E ) γ = span { s α p r ( µ ) s ∗ β : | α | = | β | , α, β, µ ∈ L ∗ ( E ) } . Moreover, since T is a compact group, there exists a faithful conditionalexpectation Ψ : C ∗ ( E, L , E ) → C ∗ ( E, L , E ) γ . The Thue–Morse sequence.
We interchangeably use the terms ‘block’ or‘word’ with the term ‘path’. Given an infinite path x = . . . x − x − .x x x . . . or afinite path x = x x . . . x | x | , we write x [ m,n ] for the block x m x m +1 . . . x n for m < n .Recall that the (two-sided) Thue–Morse sequence ω = · · · ω − ω − .ω ω ω · · · is defined by ω = 0, ω = ¯0 := 1 (¯1 := 0), ω [0 , := ω ω ¯ ω ¯ ω = 0110, ω [0 , := ω ω ω ω ¯ ω ¯ ω ¯ ω ¯ ω = 01101001 and so on, and then ω − i := ω i − for i ≥
1. Let( E Z , L ) be the following labeled graph labeled by the Thue–Morse sequence: · · · · · ·• • • • • • • • • (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) ω = 0 ω = 1 1 0 v v v − v − v − v − v v v For a path α = α . . . α | α | ∈ L ∗ ( E Z ), we also write α − := α | α | . . . α and α := α . . . α | α | . Then clearly ( α − ) − = α , α − = ( α ) − , and ( αβ ) − = β − α − for α, β ∈ L ∗ ( E Z ).2.3. Labeled graph C ∗ -algebra associated to the Thue–Morse sequence. From now on, we mainly consider the labeled space ( E Z , L , E Z ) and the labeledgraph C ∗ -algebra C ∗ ( E Z , L , E Z ) associated to the Thue–Morse sequence. Clearlythe C ∗ -algebra C ∗ ( s a , p A ) := C ∗ ( E, L , E ) is unital with the unit s s ∗ + s s ∗ and isknown to be a simple C ∗ -algebra which is finite but not AF (see [11, Example 4.9]and [12, Theorem 3.7]).It is useful to note that E Z = (cid:8) (cid:71) ≤ i ≤ K | α i | = m r ( α i ) : m ≥ , K ≥ (cid:9) . (2)Actually since the graph E Z consists of a single bi-infinite path, it is easy to see thatfor each v ∈ E , L ( E l v ) = { α } is a singleton set and so [ v ] l = r ( α ). Equivalently, r ( α ) = [ v ] | α | for any v ∈ r ( α ) . Thus every set A ∈ E Z is a disjoint union of r ( α i ) for i = 1 , . . . , n by (1). Moreover,we can assume that the lengths | α i | are all equal since r ( α ) = r (0 α ) (cid:116) r (1 α ). Hence(2) follows.Furthermore for α, β ∈ L ∗ ( E Z ) with | α | = | β | , the intersection of two rangesets r ( α ) ∩ r ( β ) is nonempty if and only if α = β . That is, s α p r ( µ ) s ∗ β (cid:54) = 0 for RACE ON THE THUE–MORSE LABELED GRAPH C ∗ -ALGEBRA 5 | α | = | β | < | µ | if and only if α = β and ∅ (cid:54) = r ( µ ) ⊂ r ( α ). Note that r ( µ ) ⊂ r ( α ) ifand only if µ = βα for some β ∈ L ∗ ( E Z ). From this observation, we have C ∗ ( E Z , L , E Z ) γ = span { s α p r ( µ ) s ∗ β : | α | = | β | , α, β, µ ∈ L ∗ ( E Z ) } = span { s α p r ( µ ) s ∗ α : α, µ ∈ L ∗ ( E Z ) } = span { s α p r ( βα ) s ∗ α : α, β ∈ L ∗ ( E Z ) } . Remark . We will use in the Section 4 the following structural properties of theAF algebra C ∗ ( E Z , L , E Z ) γ .(i) For each k ≥
1, let F k = span { s α p r ( βα ) s α : | α | = | β | = k } , be the finite dimensional C ∗ -subalgebra of C ∗ ( E Z , L , E Z ) γ and let ι k : F k → F k +1 be the inclusion map: ι k ( s α p r ( βα ) s α ) = (cid:88) a ∈{ , } s αa p r ( βαa ) s αa = (cid:88) a,b ∈{ , } s αa p r ( bβαa ) s αa . (3)Then the AF algebra C ∗ ( E Z , L , E Z ) γ is the inductive limit, lim −→ ( F k , ι k ).(ii) For each k ≥
1, the C ∗ -subalgebra F k of C ∗ ( E Z , L , E Z ) γ is commutative andisomorphic to C d k where d k = |L ( E k ) | . This directly follows from the factthat the product of two elements s α p r ( βα ) s ∗ α , s α (cid:48) p r ( β (cid:48) α (cid:48) ) s ∗ α (cid:48) ∈ F k is equal to( s α p r ( βα ) s ∗ α )( s α (cid:48) p r ( β (cid:48) α (cid:48) ) s ∗ α (cid:48) ) = δ α,α (cid:48) s α p r ( βα ) ∩ r ( β (cid:48) α (cid:48) ) s ∗ α (cid:48) = δ βα,β (cid:48) α (cid:48) s α p r ( βα ) s ∗ α . Labeled paths on the Thue–Morse sequence
For b, c ∈ L ∗ ( E Z ) with | c | = n , the product b × c denotes the block (of length | b | × | c | )obtained by n copies of b or b according to the rule: choosing the i th copy as b if c i = 0 and b if c i = 1 (see [14]). For example, if b = 01 and c = 011, then b × c isequal to bbb = 011010. Then the recurrent sequence01 × × · · · is a one-sided Morse sequence.To compute the trace values on C ∗ ( E Z , L , E Z ), we will use the following notationfor convenience. Let 0 (0) := 0, 1 (0) := 1 and0 ( n ) := 0 ( n − ( n − , ( n ) := 1 ( n − ( n − for all n , inductively. Then the length of i ( n ) ( i = 0 or 1) is 2 n , and0 ( n ) = 01 × · · · × (cid:124) (cid:123)(cid:122) (cid:125) ( n − -times ×
01 and 1 ( n ) = 01 × · · · × (cid:124) (cid:123)(cid:122) (cid:125) ( n − -times × . Note that i ( n ) = i ( n ) = (1 − i ) ( n ) and( i ( n ) ) − = (cid:40) i ( n ) , if n is odd, i ( n ) , if n is even S. H. KIM for i = 0 ,
1. Using the notation, we see that this Morse sequence has a fractalaspect: ω = · · · ω − ω − ω − ω − .ω ω ω ω · · · = · · · . · · · = · · · (1) (1) (1) (1) . (1) (1) (1) (1) · · · = · · · (2) (2) (2) (2) . (2) (2) (2) (2) · · · ...= (cid:26) · · · ( n ) ( n ) ( n ) ( n ) . ( n ) ( n ) ( n ) ( n ) · · · , if n is odd, · · · ( n ) ( n ) ( n ) ( n ) . ( n ) ( n ) ( n ) ( n ) · · · , if n is even. (4) Lemma 3.1.
Let α be a labeled path in L ∗ ( E Z ) . Then both α − and α appear in L ∗ ( E Z ) . Furthermore, r ( α ) is always an infinite set.Proof. If α ∈ L ∗ ( E Z ), then we can find n, m ∈ Z with n < m and α = ω [ n,m ] . Thus α − = ω [ − m − , − n − belongs to L ∗ ( E Z ). Moreover we can choose a large l > α = ω [ n,m ] is a subpath of ω [ − · l , · l − = 1 (2 l ) (2 l ) (2 l ) (2 l ) = 1 (2 l +1) . Then since ω [ − · l , · l − = 1 (2 l +1) = 0 (2 l +1) , the labeled path α is in L ∗ ( E Z ). Moreover 1 (2 l +1) appears infinitely many times(see (4)), so that r ( α ) is infinite. (cid:3) Remark . We emphasize from [9] that the (two-sided) Thue–Morse sequence is overlap-free : for any β ∈ L ∗ ( E Z ), the labeled path of the form βββ [1 ,n ] with n ≤ | β | does not appear in L ∗ ( E Z ). Thus s βββ [1 ,n ] = p r ( βββ [1 ,n ] ) = 0. For example, neither1001000 nor 1001001 = (100)(100)1 appear in the Thue–Morse sequence, and hence100100 / ∈ L ∗ ( E Z ). Lemma 3.3. If α has length n and i ( n )1 i ( n )2 α ∈ L ∗ ( E Z ) ( or α i ( n )1 i ( n )2 ∈ L ∗ ( E Z )) for i , i = 0 , , then α = i ( n ) for i = 0 or .Proof. We use an induction on n . When n = 0 or 1, it is easy to see that theassertion is true. Now we assume that the assertion holds for all n ≤ N ( N ≥ β = i ( N +1)1 i ( N +1)2 α ∈ L ∗ ( E Z )with | α | = 2 N +1 . Then β can be written as β = i ( N +1)1 i ( N +1)2 α = i ( N )1 i N ) i ( N )2 i N ) α α with | α | = | α | = 2 N . From the induction hypothesis, α α must be of the form j ( N )1 j ( N )2 . Thus it is enough to show that α α is neither 0 ( N ) ( N ) nor 1 ( N ) ( N ) .Suppose that α α = 0 ( N ) ( N ) . Since 0 ( N ) ( N ) ( N ) / ∈ L ∗ ( E Z ), the block i ( N )2 i N ) must be 0 ( N ) ( N ) . If i ( N )1 i N ) = 0 ( N ) ( N ) , then β = 0 ( N ) ( N ) ( N ) ( N ) ( N ) ( N )RACE ON THE THUE–MORSE LABELED GRAPH C ∗ -ALGEBRA 7 and such β can not be contained in L ∗ ( E Z ) (see Remark 3.2). If i ( N )1 i N ) =1 ( N ) ( N ) , we have β = 1 ( N ) ( N ) ( N ) ( N ) ( N ) ( N ) , but then β / ∈ L ∗ ( E Z ) again. Therefore we obtain that α α (cid:54) = 0 ( N ) ( N ) and itfollows that α α (cid:54) = 1 ( N ) ( N ) by considering β . Hence α is either 0 ( N +1) or 1 ( N +1) .Finally, the case α i ( n )1 i ( n )2 can be done from ( α i ( n )1 i ( n )2 ) − = j ( n )1 j ( n )2 α − ∈ L ∗ ( E Z ). (cid:3) Lemma 3.3 implies that if we choose a long labeled path (long enough to con-tain at least two consecutive i ( n ) -blocks), then its succeeding and preceding la-beled paths must have certain forms. This is no longer true in general, for ex-ample 0 ( n ) α ∈ L ∗ ( E Z ) with | α | = 2 n does not imply α = 0 ( n ) or 1 ( n ) because0 ( n − ( n − ( n − ( n − = 0 ( n ) α ∈ L ∗ ( E Z ).One can observe from the definition of the Morse sequence that ω [2 k, k +1] isneither 00 nor 11. Also any labeled path of length 5 must contain 00 or 11. Thus if α = α . . . α | α | is a path of length | α | ≥
5, then we can actually figure out whether α = ω k or not (namely, α = ω k +1 ) for some k . Therefore if α = α . . . α , then α can be uniquely written in the form using i (1) -blocks. For example, α = 01101then α = (01)(10)1 = 0 (1) (1) α = 0(11)(01) (cid:54) = 0 i (1)1 i (1)2 for any i , i ∈ { , } . We can extend this fact tothe i ( n ) -blocks for n ≥ Lemma 3.4.
Let α ∈ L ∗ ( E Z ) be a path of the form i n ) i n ) i n ) i n ) i n ) for some n > . Then α is written as one and only one of the following forms: i n +1) i n +1) i n ) or i n ) i n +1) i n +1) . (5) Proof.
The proof directly follows by substituting i ( n ) for i . (cid:3) By Lemma 3.3 and Lemma 3.4, we can write any labeled path in L ∗ ( E Z ) using i ( n ) -blocks. Proposition 3.5.
Let α be a labeled path with length | α | ≥ . Then there exists n ≥ such that α = γ i ( n )1 . . . i ( n ) k γ , ≤ k ≤ , (6) where γ is a final subpath of ( n ) or ( n ) , and γ is an initial subpath of ( n ) or ( n ) with length ≤ | γ | , | γ | < n ( n is not necessarily unique ) . Moreover, if we fixsuch an n , the expression of α in i ( n ) -blocks is unique.Proof. For 2 ≤ | α | ≤
4, let n = 0. If | α | = 5, Lemma 3.4 shows that n = 1 is thedesired one.Now we assume that 6 · n ≤ | α | ≤ · n +1 for n ≥
0. Since | i ( n ) | = 2 n and6 ≤ | α | / | i ( n ) | ≤ α contains at least 5 and at most 12 of i ( n ) -blocks. Thus α canbe written as α = µ i ( n )1 . . . i ( n ) k µ , ≤ k ≤ . S. H. KIM
Applying Lemma 3.3 and Lemma 3.4, we can reduce the number of i ( n ) -blocks tohave α = µ (cid:48) j ( n +1)1 . . . j ( n +1) l µ (cid:48) , ≤ l ≤ . If l ≥
5, then we apply lemmas again and we see that α contains at least two andat most four blocks of 0 ( n ) , 1 ( n ) . Lemma 3.3 implies that γ and γ are subpaths of0 ( n ) , 1 ( n ) .We now prove the uniqueness of the expression. If α has two different expressionsin i ( n ) -blocks as in (6) for n ≥
2, then by Lemma 3.4, α must have two differentexpressions in i ( n − -blocks. By induction, we conclude that α can be written astwo different forms in i (1) -blocks. This contradicts to the fact that any labeled path α with | α | ≥ i (1) -blocks isunique. (cid:3) Example 3.6.
Let α = 00101101001011001101001100101100 be a labeled path in L ∗ ( E Z ). Note that α [1 , = 00 is not equal to 0 (1) or 1 (1) . Thus by Lemma 3.4, α can be written in the following form using i (1) -blocks α = 00101101001011001101001100101100= 0 / / / / / / / / / / / / / / / /
0= 0 / (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) / . (Note here that two consecutive i ( n )1 i ( n )2 blocks between ‘/’ form a i ( n +1)1 block.)Applying Lemma 3.4 repeatedly, we obtain α = 0 / (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) /
0= 00 (1) / (1) (1) / (1) (1) / (1) (1) / (1) (1) / (1) (1) / (1) (1) / (1) (1) /
0= 00 (1) / (2) (2) (2) (2) (2) (2) (2) /
0= 00 (1) (2) / (2) (2) / (2) (2) / (2) (2) /
0= 00 (1) (2) / (3) (3) (3) / γ (3) (3) (3) γ , where γ = 00 (1) (2) = 0010010 is a final subpath of 1 (3) = 10010110, and γ = 0is an initial subpath of 0 (3) = 01101001. If αβ ∈ L ∗ ( E Z ) for some labeled path β of length 2 −
1, then by Lemma 3.3, γ β = 0 β = i (3) for some i = 0 or 1. Thus0 β = 0 (3) . Similarly, if β α ∈ L ∗ ( E Z ) with | β | = 1 then β γ = β (1) (2) = 1 (3) ,and we have β = 1. Thus β αβ = 1 (3) (3) (3) (3) (3) . Since ω [0 , = 0 (3) (3) (3) (3) (3) (3) (3) (3) , we now see that β αβ = ω [8 , and α = ω [9 , . Notation . We define A m α A n to be the set A m α A n := { β αβ ∈ L ∗ ( E Z ) : | β | = m, | β | = n } for m, n ≥
0. We identify A m α with A m α A and α A n with A α A n . RACE ON THE THUE–MORSE LABELED GRAPH C ∗ -ALGEBRA 9 We finish this section by proving the following lemma which is necessary in thenext section to show that C ∗ ( E Z , L , E Z ) has a trace. Lemma 3.8.
Let α be a labeled path with length | α | ≥ . Then the cardinality ofthe set A α A is , , or .Proof. We show that |A α A| ≥ |A α A| = 4. If 0 α / ∈ L ∗ ( E Z ), then A α A ⊂{ α , α } and |A α A| ≤
2. Similar arguments with the cases 1 α / ∈ L ∗ ( E Z ), α / ∈L ∗ ( E Z ), or α / ∈ L ∗ ( E Z ) show that if |A α A| ≥
3, then { α, α, α , α } ⊂ L ∗ ( E Z ).We know from Proposition 3.5 that the path α can be written as α = γ i ( n )1 . . . i ( n ) k γ , ≤ k ≤ n ≥
0. If 0 < | γ | < n , Proposition 3.5 also shows that γ j ( j = 0 or 1)is an initial subpath of i ( n ) for some i = 0 or 1, and thus j is uniquely determinedby γ . This implies that |A α A| ≤
2. Hence when |A α A| ≥ γ should be anempty word and similarly, so is γ . Thus if |A α A| ≥
3, we can write α = i n ) . . . i k ( n ) , ≤ k ≤ , for some n . It is an easy but time-consuming work to show that there exist onlyfour labeled paths α ( n ) ( n ) , ( n ) ( n ) , ( n ) ( n ) ( n ) ( n ) , ( n ) ( n ) ( n ) ( n ) satisfying |A α A| ≥
3, and in each case one can show that |A α A| = 4 by a straight-forward computation. (cid:3) The trace on the labeled graph C ∗ -algebra C ∗ ( E Z , L , E Z )Note that if τ : C ∗ ( E Z , L , E Z ) → C is a tracial state, it satisfies τ ( s α p r ( βα ) s ∗ α ) = τ ( p r ( βα ) ) ,τ ( p r ( α ) ) = τ ( s p r ( α s ∗ ) + τ ( s p r ( α s ∗ ) = τ ( p r ( α ) + τ ( p r ( α )for p r ( α ) , s α p r ( βα ) s ∗ α ∈ C ∗ ( E Z , L , E Z ) γ . Let χ := span { p A : A ∈ E Z } be the commutative C ∗ -subalgebra of C ∗ ( E Z , L , E Z ) γ generated by projections p A , A ∈ E Z . The following lemma shows that every faithful state on C ∗ ( E Z , L , E Z ) γ (oron χ ) with the above property extends to a tracial state on C ∗ ( E Z , L , E Z ). By S ( D )we denote the set of all faithful tracial states on a C ∗ -algebra D . Lemma 4.1.
Let S := S ( C ∗ ( E Z , L , E Z )) be the set of tracial states of C ∗ ( E Z , L , E Z ) and let S γ := { τ ∈ S ( C ∗ ( E Z , L , E Z ) γ ) : τ ( s α p r ( βα ) s ∗ α ) = τ ( p r ( βα ) ) } , S χ := { τ ∈ S ( χ ) : τ ( p r ( α ) ) = τ ( p r ( α ) + τ ( p r ( α ) } . Then the restriction maps φ : S → S γ and φ : S γ → S χ are bijections with theinverses φ − ( τ ) = τ ◦ Ψ for τ ∈ S γ and φ − ( τ )( s α p r ( βα ) s ∗ α ) = τ ( p r ( βα ) ) for τ ∈ S χ , respectively. Proof.
To prove that φ is injective, it is enough to show that τ = τ ◦ Ψ for all τ ∈ S . Let s α p A s ∗ β ∈ C ∗ ( E Z , L , E Z ) with | α | (cid:54) = | β | . Without loss of generality, we mayassume that | α | > | β | . Since τ is a trace, we have τ ( s α p A s ∗ β ) = τ ( s ∗ β s α p A ) = (cid:26) τ ( p r ( β ) s α p A ) = τ ( s α p A ) if α = βα , τ ( s α p A ) = τ ( p A s α ) = τ ( s α p r ( A,α ) ) = · · · = τ ( s α p r ( A,α ) ) = 0 , because α / ∈ L ∗ ( E Z ). Thus τ ≡ C ∗ ( E Z , L , E Z ) \ C ∗ ( E Z , L , E Z ) γ and hence wehave τ = τ ◦ Ψ.If we prove that τ ◦ Ψ ∈ S for all τ ∈ S γ , then φ is surjective. By the continuityof τ and Ψ, it is enough to show that τ (Ψ( XY )) = τ (Ψ( Y X )) (7)for
X, Y ∈ span { s α p A s ∗ β : α, β ∈ L ∗ ( E ) , A ⊂ r ( α ) ∩ r ( β ) } . Since τ and Ψ islinear, we are reduced to proving (7) for X = s α p A s ∗ β , Y = s µ p B s ∗ ν . Moreover, theequality s α p A s ∗ β = (cid:80) δ ∈L ( AE n ) s αδ p r ( A,δ ) s ∗ βδ yields that we only need to prove (7)when | β | = | µ | . Note from γ z ( XY ) = z | α |−| β | + | µ |−| ν | XY and γ z ( Y X ) = z −| α | + | β |−| µ | + | ν | Y X that | α | − | β | + | µ | − | ν | = | α | − | ν | (cid:54) = 0 = ⇒ Ψ( XY ) = Ψ( Y X ) = 0 . Hence we may assume that | α | = | ν | and | β | = | δ | . Then we obtain that τ (Ψ( XY )) = τ ( XY ) = τ ( s α p A s ∗ β s µ p B s ∗ ν )= τ ( δ β,µ s α p A ∩ B s ∗ ν ) = τ ( δ β,µ δ α,ν s α p A ∩ B s ∗ α )= τ ( δ β,µ δ α,ν p A ∩ B ) ,τ (Ψ( Y X )) = τ ( δ β,µ δ α,ν p B ∩ A )= τ (Ψ( XY )) , and φ is bijective.Next, we prove that φ is bijective. Let τ ∈ S χ . Recall from Remark 2.4 (i) that C ∗ ( E Z , L , E Z ) γ = lim −→ ( F k , ι k ) where F k = span { s α p r ( βα ) s ∗ α : α, β ∈ L ∗ ( E Z ) with | α | = | β | = k } ∼ = C d k with d k = |L ∗ ( E k Z ) | . Define τ k : F k → C by τ k ( s α p r ( βα ) s ∗ α ) := τ ( p r ( βα ) ). Then τ k is a state on F k for each k ≥
1. Also for the inclusion ι k : F k → F k +1 , we have RACE ON THE THUE–MORSE LABELED GRAPH C ∗ -ALGEBRA 11 τ k ( x ) = τ k +1 ( ι k ( x )) for x ∈ F k and k ≥
1. In fact, τ k +1 ( ι k ( s α p r ( βα ) s ∗ α )) = τ k +1 ( (cid:88) a ∈{ , } s αa p r ( βαa ) s ∗ αa )= (cid:88) a ∈{ , } τ ( p r ( βαa ) ) = τ ( p r ( βα ) )= τ k ( s α p r ( βα ) s ∗ α ) . Thus τ := lim −→ τ k extends to a state on C ∗ ( E Z , L , E Z ) γ . Moreover τ ( s α p r ( βα ) s ∗ α ) = τ ( p r ( βα ) ) = τ ( p r ( βα ) ), and hence τ ∈ S γ . It is now obvious that φ is bijective. (cid:3) Remark . Let τ ∈ S χ . The equality r ( α ) = (cid:70) µα ∈A m α r ( µα ) implies that τ ( p r ( α ) ) = (cid:88) µα ∈A m α τ ( p r ( µα ) ) . In addition, p r ( α ) = (cid:80) αµ ∈ α A n s µ p r ( αµ ) s ∗ µ implies that τ ( p r ( α ) ) = (cid:88) αµ ∈ α A n τ ( p r ( αµ ) ) . Combining these two, we obtain τ ( p r ( α ) ) = (cid:88) µ ∈A m α A n τ ( p r ( µ ) )for any m, n ≥
0. For example, let α = 0101. Since A α = { } and A α A = { } , we have τ ( p r (0101) ) = τ ( p r (100101) ) = τ ( p r (10010110) ) . By (2), any linear functional on χ = span { P A : A ∈ E Z } is determined byits values on the projections p r ( α ) , α ∈ L ∗ ( E Z ). Note that χ is a commutative C ∗ -algebra which is the inductive limit of the finite dimensional C ∗ -subalgebras χ k := span { P r ( α ) : | α | = k } ∼ = C |L ( E k ) | . Thus any continuous linear functional ψ : ∪ k ≥ χ k → C naturally extends to a linear functional on χ . Let φ be a linearfunctional on ∪ k ≥ χ k given by(i) φ ( p r ( α ) ) = 1 / α with | α | = 3,(ii) for α with | α | ≥ φ ( p r ( iα ) ) = (cid:26) φ ( p r ( α ) ) , if iα / ∈ L ∗ ( E Z ) ,φ ( p r ( α ) ) / , if iα ∈ L ∗ ( E Z ) . Then φ is positive and of norm 1, hence it extends to a state on χ . Note that (ii)holds for all paths α ∈ L ∗ ( E Z ) except α = 0 ,
1. The paths 0 , α such that |A α A| = 3. Theorem 4.3.
The state φ on χ extends to a trace ˜ φ on C ∗ ( E Z , L , E Z ) by ˜ φ ( s α p A s ∗ β ) = δ α,β φ ( p A ) for A ⊂ r ( α ) ∩ r ( β ) . In particular, ˜ φ ( p r ( α ) ) = , if α = 00 , , , if α = 01 , , , if | α | = 3 , and ˜ φ ( p r ( α ) ) = 2 ˜ φ ( p r (0 α ) ) = 2 ˜ φ ( p r (1 α ) ) for α ∈ L ( E ≥ ) with α, α ∈ L ( E ≥ ) .Proof. We will prove that the state φ on χ belongs to S χ . Then the assertion followsfrom Lemma 4.1. For this we first show that for any labeled path α , φ ( p r ( α ) ) = φ ( p r ( α − ) ) . We use the induction on the length of α . From the definition of φ , it is obvious that φ ( p r ( α ) ) = φ ( p r ( α − ) ) for | α | ≤
3. We now assume that this holds for all labeledpath α with length | α | ≤ N for some N ≥
3. Let j βj ∈ L ∗ ( E Z ) for some j , j = 0or 1, and | β | = N −
1. By Lemma 3.8, we know that A β A is one of the following:(i) { j βj } ,(ii) { j βj , j βj } ,(iii) { j βj , j βj } ,(iv) { j βj , j βj } , and(v) { j βj , j βj , j βj , j βj ) } .Also for each case, we can show that φ ( p r ( j βj ) ) = φ ( p r ( j β − j ) ). Actually, if A β A = { j βj } as in (i), first note that j β, βj / ∈ L ∗ ( E Z ) and so β − j , j β − / ∈L ∗ ( E Z ). Thus φ ( p r ( j βj ) ) = φ ( p r ( βj ) ) = φ ( p r ( j β − ) )= φ ( p r ( β − ) ) = φ ( p r ( β ) )= φ ( p r ( j β ) ) = φ ( p r ( β − j ) )= φ ( p r ( j β − j ) ) . In case (ii), we have φ ( p r ( j βj ) ) = φ ( p r ( βj ) ) / φ ( p r ( j β − ) ) / φ ( p r ( β − ) ) / φ ( p r ( β ) ) / φ ( p r ( j β ) ) = φ ( p r ( β − j ) )= φ ( p r ( j β − j ) ) . The rest cases can be shown similarly. Then from the above observation, we have φ ( p r ( α ) ) = φ ( p r ( α − ) ) = φ ( p r (0 α − ) ) + φ ( p r (1 α − ) ) = φ ( p r ( α ) + φ ( p r ( α ) , which shows that φ ∈ S χ . This completes the proof. (cid:3) Proposition 4.4.
The trace ˜ φ on the C ∗ -algebra C ∗ ( E Z , L , E Z ) is a unique trace.Proof. By Lemma 4.1 and Theorem 4.3, it is enough to show that |S χ | ≤
1. Let τ ∈ S χ and set b n, := τ ( p r (0 ( n ) ( n ) ) ) , b n, := τ ( p r (0 ( n ) ( n ) ) ) ,b n, := τ ( p r (1 ( n ) ( n ) ) ) , b n, := τ ( p r (1 ( n ) ( n ) ) ) RACE ON THE THUE–MORSE LABELED GRAPH C ∗ -ALGEBRA 13 for n ≥
0. (If two states on χ have the common values at p r ( i ( n ) j ( n ) ) for all i, j = 0 , n ≥
0, then they are the same state.) We first show b n, = b n, < b n, = b n, .This particularly gives b , + b , = . By Lemma 3.3, we have b n, = τ ( p r (0 ( n ) ( n ) ) ) = τ ( p r (0 ( n ) ( n ) ( n ) ) ) + τ ( p r (0 ( n ) ( n ) ( n ) ) )= τ ( p r (0 ( n ) ( n ) ( n ) ) ) + τ ( p r (1 ( n ) ( n ) ) )= τ ( p r (0 ( n ) ( n ) ( n ) ) ) + b n, , (8) b n, = τ ( p r (1 ( n ) ( n ) ) ) = τ ( p r (0 ( n ) ( n ) ( n ) ) ) + τ ( p r (1 ( n ) ( n ) ( n ) ) )= τ ( p r (0 ( n ) ( n ) ( n ) ) ) + τ ( p r (1 ( n ) ( n ) ) )= τ ( p r (0 ( n ) ( n ) ( n ) ) ) + b n, , and thus we have b n, − b n, = b n, − b n, = τ ( p r (0 ( n ) ( n ) ( n ) ) ) >
0, which shows b n, = b n, > b n, . Similarly b n, > b n, can be shown, and from b n, = τ ( p r (0 ( n ) ( n ) ) ) = τ ( p r (1 ( n ) ( n ) ( n ) ( n ) ) ) = τ ( p r (1 ( n +1) ( n +1) ) ) = b n +1 , ,b n, = τ ( p r (1 ( n ) ( n ) ) ) = τ ( p r (0 ( n ) ( n ) ( n ) ( n ) ) ) = τ ( p r (0 ( n +1) ( n +1) ) ) = b n +1 , , we also see that b n, = b n +1 , = b n +1 , = b n, . Thus the numbers b n,k for n ≥ k = 1 , , , b , is given.Now (8) gives us b n, = τ ( p r (0 ( n ) ( n ) ( n ) ) ) + b n, = τ ( p r (0 ( n ) ( n ) ( n ) ( n ) ) ) + τ ( p r (1 ( n ) ( n ) ( n ) ( n ) ) ) + b n, = τ ( p r (1 ( n ) ( n ) ( n ) ( n ) ( n ) ) ) + τ ( p r (1 ( n +1) ( n +1) ) ) + b n, = τ ( p r (1 ( n +1) ( n +1) ( n +1) ) ) + b n +1 , + b n, = 2 b n +1 , + b n, , and so (cid:18) b n, b n, (cid:19) = (cid:18) (cid:19) (cid:18) b n +1 , b n +1 , (cid:19) . Since b , + b , = , if we put t = b , , (cid:18) b n, b n, (cid:19) = (cid:18) − / /
21 0 (cid:19) n (cid:18) b , b , (cid:19) = (cid:18)(cid:18) / √ / √ / √ − / √ (cid:19) (cid:18) / − (cid:19) (cid:18) / √ / √ / √ − / √ (cid:19)(cid:19) n (cid:18) b , b , (cid:19) = 13 (cid:18) − (cid:19) (cid:18) / n
00 ( − n (cid:19) (cid:18) − (cid:19) (cid:18) t / − t (cid:19) = 13 (cid:18) (1 / n +1 + (3 t − / − n (1 / n − (3 t − / − n (cid:19) . (9)The state τ is faithful, so b n, and b n, must be strictly positive. Thus t = 1 /
6, andhence τ = φ . (cid:3) Remark . By (9), the unique trace ˜ φ on C ∗ ( E Z , L , E Z ) satisfies˜ φ ( p r (0 ( n ) ( n ) ) ) = ˜ φ ( p r (1 ( n ) ( n ) ) ) = 1 / (6 · n ), and˜ φ ( p r (0 ( n ) ( n ) ) ) = ˜ φ ( p r (1 ( n ) ( n ) ) ) = 1 / (3 · n ) . (10)By Proposition 3.5, every labeled path α with | α | ≥ α = γ i ( n )1 . . . i ( n ) k γ for 2 ≤ k ≤
4. We thus can compute ˜ φ ( p r ( α ) ) for all labeledpaths α using (10). For example, α = x [10 , is written as α = 010 (2) (2) (2) (2) (2) = 010 (2) (3) (3) , and then ˜ φ ( p r ( α ) ) = ˜ φ ( p r (1 (3) (3) (3) ) ) = ˜ φ ( p r (1 (3) (3) ) ) = 1 / (6 · ).5. K -groups of C ∗ ( E Z , L , E Z )Note that for i ( n ) -blocks with a even number n , the first and last alphabets of i ( n ) areexactly i . For example, 0 (2) = ˙011 ˙0 but 0 (3) = ˙0110100 ˙1. Therefore it is convenientto present the preceding alphabet of labeled paths: A (2) (2) = { (2) (2) (2) } and A (2) (2) = { (2) (2) } , while A (3) (3) = { (3) (3) (3) } and A (3) (3) = { (3) (3) } .We first briefly review the K-groups of labeled graph C ∗ -algebras. We let Ω l = { [ v ] l : v ∈ E } , and Ω = ∪ l ≥ Ω l be the set of all generalized vertices. By Z (Ω) wedenote the additive group span Z { χ [ v ] l : [ v ] l ∈ Ω } .Let (I − Φ) : Z (Ω) → Z (Ω) be the linear map defined by(I − Φ)( χ [ v ] l ) = χ [ v ] l − (cid:88) a ∈A χ r ([ v ] l ,a ) for [ v ] l ∈ Ω . Then the following is known in [1, Corollary 8.3].(i) K ( C ∗ ( E, L , E )) ∼ = ker(I − Φ),(ii) K ( C ∗ ( E, L , E )) ∼ = coker(I − Φ) via the map [ p [ v ] l ] (cid:55)→ χ [ v ] l + Im(I − Φ).As shown in [12], C ∗ ( E Z , L , E Z ) is isomorphic to the crossed product C ( X ) (cid:111) σ Z of a two-sided Thue–Morse subshift ( X, σ ), and hence K ( C ∗ ( E Z , L , E Z )) ∼ = Z follows from [10]. To compute K -group of C ∗ ( E Z , L , E Z ), first note that (cid:88) a ∈A p r ( aα ) = p r ( α ) = (cid:88) a ∈A s a p r ( αa ) s ∗ a yields that (cid:88) a ∈A [ p r ( aα ) ] = [ p r ( α ) ] = (cid:88) a ∈A [ p r ( αa ) ] . (11)Combining (11) and Proposition 3.5, we can rewrite [ p r ( α ) ] as (cid:88) k i [ p r ( β i ) ] where k i ∈ N and β i ∈ { ( n ) ( n ) ( n ) , ( n ) ( n ) ( n ) , . . . , ( n ) ( n ) ( n ) } for some n ≥ p β i ]’s. RACE ON THE THUE–MORSE LABELED GRAPH C ∗ -ALGEBRA 15 Lemma 5.1.
For all n ≥ , we have [ p r (0 ( n ) ( n ) ( n ) ) ] = [ p r (1 ( n ) ( n ) ( n ) ) ] , (12)[ p r (0 ( n ) ( n ) ( n ) ) ] = [ p r (1 ( n ) ( n ) ( n ) ) ] = [ p r (0 ( n ) ( n ) ( n ) ) ] = [ p r (1 ( n ) ( n ) ( n ) ) ] . (13) Moreover, if we let a n := [ p r (0 ( n ) ( n ) ( n ) ) ] and b n := [ p r (0 ( n ) ( n ) ( n ) ) ] , (14) then we have a n = 2 b n +1 , b n = a n +1 + b n +1 , and a n (cid:54) = b n .Proof. First notice that the equality (11) yields that[ p r (1 ( n ) ( n ) ( n ) ) ] = [ p r (0 ( n ) ( n ) ) ] = [ p r (0 ( n ) ( n ) ( n ) ] and [ p r (0 ( n ) ( n ) ( n ) ) ] = [ p r (1 ( n ) ( n ) ) ] = [ p r (1 ( n ) ( n ) ( n ) ] . The equality[ p r (0 ( n ) ( n ) ) ] = [ p r (1 ( n ) ( n ) ( n ) ( n ) ) ] = [ p r (1 ( n +1) ( n +1) ) ] = [ p r (1 ( n +1) ( n +1) ( n +1) ) ] + [ p r (1 ( n +1) ( n +1) ( n +1) ) ] (15)= [ p r (0 ( n +1) ( n +1) ( n +1) ) ] + [ p r (1 ( n +1) ( n +1) ( n +1) ) ] = [ p r (0 ( n +1) ( n +1) ] = [ p r (1 ( n ) ( n ) ) ] shows that (13) holds. From this, we obtain that[ p r (0 ( n ) ( n ) ( n ) ) ] = [ p r (0 ( n ) ( n ) ( n ) ( n ) ) ] + [ p r (0 ( n ) ( n ) ( n ) ( n ) ) ] = [ p r (1 ( n ) ( n ) ( n ) ( n ) ( n ) ( n ) ) ] + [ p r (0 ( n +1) ( n +1) ) ] (16)= [ p r (1 ( n +1) ( n +1) ( n +1) ) ] + [ p r (0 ( n +1) ( n +1) ) ] = 2[ p r (0 ( n +1) ( n +1) ) ] , and similarly [ p r (1 ( n ) ( n ) ( n ) ) ] = 2[ p r (0 ( n +1) ( n +1) ) ] , so (12) holds.We already obtain from (15) that b n = a n +1 + b n +1 and from (16) that a n = 2 b n +1 .Combining these we get a n − b n = − ( a n +1 − b n +1 )which then implies that a n − b n does not belong to Im(I − Φ), and hence a n (cid:54) = b n . (cid:3) As a result, we have the following.
Proposition 5.2.
Let a n , b n ( n ≥ ) be elements in K ( C ∗ ( E Z , L , E Z )) given by(14). Then the K -group of C ∗ ( E Z , L , E Z ) is K ( C ∗ ( E Z , L , E Z )) = span Z { a n , b n : n ≥ , a n = 2 b n +1 , b n = a n +1 + b n +1 }∼ = lim → (cid:18) Z , (cid:18) (cid:19) (cid:19) , where each Z is ordered by ( n , n ) ≥ ⇐⇒ n , n ≥ with an order unit [1] = 2 a + 4 b . From the Remark 4.5, we see that K ( ˜ φ )( a n ) = K ( ˜ φ )( b n ) = 1 / (6 · n )for all n ≥
0. Then the trace map ˜ φ : C ∗ ( E Z , L , E Z ) → C induces a surjective (notinjetive) group homomorphism K ( ˜ φ ) : K ( C ∗ ( E Z , L , E Z )) → Q (2 ∞ · Q (2 ∞ ·
3) is an additive subgroup of Q consisting of the fractions x/y for x ∈ Z and y = 2 n n ≥
1. It is easily seen that K ( C ∗ ( E Z , L , E Z )) / ker( K ( ˜ φ )) ∼ = K ( C ∗ ( E Z , L , E Z )) / (cid:104) a − b (cid:105) ∼ = Q (2 ∞ · . Representation of C ∗ ( E Z , L , E Z ) on (cid:96) ( Z )In this section, we introduce a representation of C ∗ ( E Z , L , E Z ) on the Hilbert space (cid:96) ( Z ). Recall that the directed graph E Z is as follows: · · · · · ·• • • • • • • • • (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) (cid:47) e − e − e − e − e e e e v v v − v − v − v − v v v For a canonical orthnormal basis { v n : n ∈ Z } for (cid:96) ( Z ), we define shift operators t , t on (cid:96) ( Z ) by t i ( v n ) = (cid:26) v n − , if L ( e n − ) = i t , t are partial isometries which have mutually orthogonal ranges, and t + t is a bilateral shift on (cid:96) ( Z ). Note that these two shift operators are not periodic. Proposition 6.1.
There exists an isomorphism
Φ : C ∗ ( E Z , L , E Z ) → C ∗ ( t , t ) such that Φ( s i ) = t i , i = 0 , .Proof. For a labeled path α ∈ L ∗ ( E ), we define a partial isometry t α by t α t α · · · t α | α | ,and a projection q r ( α ) by t ∗ α t α . An easy computation shows that { t α , q r ( α ) : α ∈L ∗ ( E ) } is a representation of ( E Z , L , E Z ). Hence there is a ∗ -homorphism Φ : C ∗ ( E Z , L , E Z ) → C ∗ ( t , t ) which maps s i (cid:55)→ t i . Obviously Φ is surjective. Since C ∗ ( E Z , L , E Z ) is simple, Φ is also injective. (cid:3) In general, the above proposition is also true for arbitrary labeled graphs over E Z . Let A be an (countable) alphabet set and let L : E Z → A be an arbitrarylabeling map. Then the shift operators { t a : a ∈ A} on (cid:96) ( Z ) given by t a ( v n ) = (cid:26) v n − , if L ( e n − ) = a, , otherwiseare well defined, and there exists an isomorphism Φ : C ∗ ( E Z , L , E ) → C ∗ ( t a : a ∈ A )such that Φ( s a ) = t a for a ∈ A . To prove this, it is enough to show that Φ is injective.Note that there exists a strongly continuous action λ : T → Aut( C ∗ ( t a : a ∈ A ))defined by λ z ( t a ) = zt a for z ∈ T . Moreover it is easy to see that λ z ◦ Φ =Φ ◦ γ z , where γ is a canonical gauge action on C ∗ ( E Z , L , E ). Therefore “The Gauge-Invariant Uniqueness Theorem ([5, Theorem 2.7])” guarantees that Φ is injective. RACE ON THE THUE–MORSE LABELED GRAPH C ∗ -ALGEBRA 17 References [1] T. Bates, T. M. Carlsen, and D. Pask, C ∗ -algebras of labeled graphs III - K -theory computations ,Ergod. Th. & Dynam. Sys., available on CJO2015. doi:10.1017/etds.2015.62.[2] T. Bates and D. Pask, C ∗ -algebras of labeled graphs , J. Operator Theory, (2007), 101–120.[3] T. Bates and D. Pask, C ∗ -algebras of labeled graphs II - simplicity results , Math. Scand. (2009), no. 2, 249–274.[4] T. Bates, D. Pask, I. Raeburn, and W. Szymanski, The C ∗ -algebras of row-finite graphs , NewYork J. Math. (2000), 307–324.[5] T. Bates, D. Pask, and P. Willis, Group actions on labeled graphs and their C ∗ -algebras , IllinoisJ. Math., (2012), 1149–1168.[6] J. Cuntz and W. Krieger, A class of C ∗ -algebras and topological Markov chains , Invent. Math. (1980), 251–268.[7] R. Exel and M. Laca, Cuntz-Krieger algebras for infinite matrices , J. Reine. Angew. Math. (1999), 119–172.[8] N. J. Fowler, M. Laca, and I. Raeburn,
The C ∗ -algebras of infinite graphs , Proc. Amer. Math.Soc. (2000), 2319–2327.[9] W. H. Gottschalk and G. A. Hedlund, A characterization of the Morse minimal set , Proc.Amer. Math. Soc., (1964), 70-74.[10] T. Giordano, I. F. Putnam, and C. F. Skau, Topological orbit equivalence and C ∗ -crossedproducts, J. reine angew. Math. (1995), 51-111.[11] J. A Jeong, E. J. Kang, and S. H. Kim, AF labeled graph C ∗ -algebras , J. Funct. Anal. (2014), 2153–2173.[12] J. A Jeong, E. J. Kang, S. H. Kim, and G. H. Park, Finite simple labeled graph C ∗ -algebrasof Cantor minimal subshifts , arXiv:1504.03455v2 [math.OA].[13] T. Katsura, P. Muhly, A. Sims, and M. Tomforde, Ultragraph C ∗ -algebras via topologicalquivers , J. Reine. Angew. Math. (2010), 135–165.[14] M. Keane, Generalized Morse sequences , Z. Wahrscheinlichkeitstheorie verw. Geb. (1968),335-353.[15] A. Kumjian, D. Pask, and I. Raeburn, Cuntz-Krieger algebras of directed graphs , Pacific J.Math. (1998), 161–174.[16] A. Kumjian, D. Pask, I. Raeburn, and J. Renault
Graphs, groupoids, and Cuntz-Kriegeralgebras , J. Funct. Anal. (1997), 505–541.[17] K. Matsumoto, On C ∗ -algebras associated with subshifts , Intern. J. Math. (1997), 457–374.[18] I. Raeburn, Graph Algebras , CBMS Regional Conference Series in Mathematics, vol. 103,Amer. Math. Soc., Providence, RI 2005.[19] I. Raeburn and W. Szymanski,
Cuntz-Krieger algebras of infinite graphs and matrices , Trans.Amer. Math. Soc. (2004), 39–59.[20] M. Tomforde, A unified approach to Exel-Laca algebras and C ∗ -algebras associated to graphs,J. Operator Theory, (2003), 345–368. BK21 Plus Mathematical Sciences Division, Seoul National University, Seoul, 151–747, Korea
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