数理解析研究所講究録
第 1996 巻 2016 年 13-22

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FINITE SIMPLE-ALGEBRAS OF LABELED SPACES$C^{*}$JA A JEONG, EUN JI KANG, SUN HO KIM, AND GI HYUN PARKABSTRACT. The C’-algebras of directed graphs are introduced in the $1990s$ and its study isextended to larger classes of -algebras in ma1ty ways, among which is the class of labeledgraph C’-algebras started by Bates and Pask. In this paper we survey some of our recent
results on finite labeled graph C’-algebras.
$C^{*}$

1. INTRODUCTION
A class of -algebras $C^{*}(E)$ associated to directed graphs $E$ was introduced in [14, 15]. CuntzKrieger algebras are now regarded as graph-algebras of finite graphs (graphs with fini ely
many vertices and edges). The graph -algebra $C^{*}(E)$ is the -algebra generated by a universal Cuntz Krieger $E$-family consisting of projections
and partial isometries
indexed by the vertex set
and the edge setof $E$ , which are subject to the relations determined by the $gra\iota>hE$ . If a graph $E$ has condition (K), a condition on the loop structure of $E$ , itis known [14] that the ideal structure of the -algebra $C^{*}(E)$ can be fully understood from thegraph $E$ itself. Also, if $C^{*}(E)$ is simple, it must be either AF or purely infinite. Cuntz algebrasand simple Cuntz Krieger algebras are standard examples of those simple purely infinite graph-algebras.
By a labeled graph, we mean a graph $E$ with a labeling map :ofonto thealphabet . If a setof vertex subsets satisfies certain conditions (see Chapter 2),
we call it an accommodating set and the triplea labeled space. With the alphabet, we have a trivial labeled space
and the trivial labeling map:
associated to a graph $E$ , where is the accommodating set of all vertex sets thatare either finite or cofinite. To each labeled spacewith some mild conditions, one canassociate a-algebra
generated by a universal family of projections
and partial isometries
that obey some relations given by the labeled space.
This is a similar but more complicated way to the construction of graph -algebras associatedwith graphs, and by construction every graph C’-algebra is the C’-algebra of the trivial labeledspace.
As for the simplicity of labeled graph C’-algebra, it is often enough to check the
structure of labeled paths in the labeled spaceinas the case of graph -algebras
which was well known back in the $1990s([2,7$While AF graph -algebras are exactly the -algebras $C^{*}(E)$ of graphs $E$ with no loops,
it is not so clear when a labeled graph -algebrais AF. We review the discussion
on this problem given in [8] in Section 3 after setting up some notation in Section 2. Then in
Section 4 we present the construction (given in [9]) of finite simple labeled graph -algebrasthat are not AF, which shows that the class of simple labeled graph -algebras is strictly larger
than the simple graph-algebras. For this construction we use generalized Morse sequences$:=\{e_{n}|s(e_{n})=$to label the uderlying graphwith the vertices
and the edges$n,$ $r(e_{n})=n+1,$, and show that the -algebras of these labeled graphs
are simple and non-AF (with non-zero ), but finite admitting unique tracial states.
$C^{*}$$C^{*}$$C^{*}$$C^{*}$$\{s_{e}\}_{e\epsilon E^{1}},$$\{p_{v}\}_{v\epsilon E^{0}}$$E^{0}$$E^{1}$$C^{*}$$C^{*}$$\mathcal{L}$$E^{1}arrow \mathcal{A}$$E^{1}$$\mathcal{B}c2^{E^{0}}$$\mathcal{A}$$(E, \mathcal{L}, \mathcal{B})$$\mathcal{A}=E^{1}$$E^{1}arrow \mathcal{A}$$\mathcal{L}_{id}:=id$$(E,\mathcal{L}_{id}, \mathcal{B})$$\mathcal{B}$$(E, \mathcal{L}, \mathcal{B})$$C^{*}$$p_{A}(A\in \mathcal{B})$$C^{*}(E,\mathcal{L}, \mathcal{B})$$s_{a}(a\in \mathcal{A})$$(E, \mathcal{L}, \mathcal{B})$$C^{*}$$(E, \mathcal{L}_{id}, \mathcal{B})$$C^{*}(E, \mathcal{L}, \mathcal{B})$$C^{*}$$(E, \mathcal{L}, \mathcal{B})$$C^{*}$$C^{*}$$C^{*}$$C^{*}(E, \mathcal{L}, \mathcal{B})$$C^{*}$$C^{*}$$C^{*}$$E_{Z}^{0}$$E_{\mathbb{Z}}$$\omega$$:=\mathbb{Z}$$C^{*}$$n\in \mathbb{Z}\}$$K_{1}$

2010 Mathematics Subject Cassification. $46L05,$ $46I_{4}55,$ $37A55.$
Key words and phrases. labeled graph -algebra, finite C’-algebra.
$C^{*}$$E_{Z}^{1}$$(E_{\mathbb{Z}}, \mathcal{L}_{\omega\rangle}\overline{\mathcal{E}}_{Z})$14
J. A. JEONG, E. J. KANG, S. H. KIM, AND G. H. PARK

2. PRELIMINARIES(directed)
2.1. Labeled spaces. For notational conventions we refer to [14], [2] and [3].
$E=(E^{0},E^{1},
r,s)$,
a countable set of edges
graphconsists of a countable set of vertices:
and the range, source maps
denotes the set of all finite paths, respectively. The
andand
for the setsof length $n(|\lambda|=n)$ . We write$v\in E^{0}$, where $r(v)=s(v)=v$ for. One can consider anmaps and naturally extend toiffor all , andinfinite pathwith the source
denotes theby
we denote the set of all infinite paths. For a vertex subset. For
by, we simply denote the set
sinks
in , ahd for, with abuse of notation, we writeand$A$$E^{0}$$r,$$E^{1}arrow E^{0}.$$s$$\lambda=\lambda_{1}\cdots\lambda_{n}$$E^{\geq n}$$E^{\leq n}$$E^{1},$$E^{n}$$\bigcup_{i=1}^{n}E^{i}$$\bigcup_{i=n}^{\infty}E^{i}$$E^{\geq 0}$

$r$

$s$

$r(\lambda_{i})=s(\lambda_{i+1})$$:=s(\lambda_{1})$$s(\lambda_{1}\lambda_{2}\lambda_{3}\cdots)$$\lambda_{1}\lambda_{2}\lambda_{3}\cdots$$A\subseteq E^{0},$$E^{\infty}$$\mathcal{B}\subset 2^{E_{0}}$$A$$A\cap E_{sink}^{0}$$\mathcal{B}\subset 2^{E_{0}}$$i$$A_{sink}$$\{A_{sink}:A\in \mathcal{B}\}$$\mathcal{B}_{sink}$$A\subseteq E_{0}$$\mathcal{B}\cap A:=\{B\in \mathcal{B}:B\subset A\}.$A labeled graphover a countable alphabet consists of a graph and a labeling map$E^{1}arrow A$(labeted) path, and will
, we call:. For. Similarly we can define an infinite labeled pathforuse notation, we call a repetition of . A labeled
for some
If a path is of the form
for(repeated -times)graph
is said to have a repeatable path if
defined byand sourceofare subsets of
all $n\geq 1$ . The range$\mathcal{L}^{*}(E)$$E$$\mathcal{A}$$(E, \mathcal{L})$$\lambda=\lambda_{1}\cdots\lambda_{n}\in E^{\geq 1}$$\mathcal{L}$$:=\mathcal{L}(\lambda_{1})\cdots \mathcal{L}(\lambda_{n})a$$\mathcal{L}(\lambda)$$:=\mathcal{L}(E^{\geq 1})$$\lambda\in E^{\infty}.$$\mathcal{L}(\lambda)$$\alpha=\beta\cdots\beta$$\alpha$$\beta\in \mathcal{L}^{*}(E)$$(E, \mathcal{L})$$\beta^{n}$$\beta$$s(\alpha)$$r(\alpha)$$\beta$$\alpha$$:=\beta\cdots\beta$$n$$\in \mathcal{L}^{*}(E)$$E^{0}$$\alpha\in \mathcal{L}^{*}(E\rangle$$r(\alpha)=\{r(\lambda):\lambda\in E^{\geq\downarrow}, \mathcal{L}(\lambda)=\alpha\},$$s(\alpha)=\{s(\lambda):\lambda\in E^{\geq 1}, \mathcal{L}(\lambda)=\alpha\}.$The relative range of$\alpha\in \mathcal{L}^{*}(E)$with respect to$A\subset 2^{E^{0}}$is defined to be$r(A, \alpha)=\{r(\lambda):\lambda\in E^{\geq 1}, \mathcal{L}(\lambda)=\alpha, s(\lambda)\in A\}.$for $1\leq i\leq j\leq|\alpha|.$by
We denote the subpathofsubpath of the form
is called an initial path of . The symbol is regarded as an initial
path of every path.
,
for allandLet
be a collection of subsets of . If
.
set
for
accommodating
We call an
is said to be closed under relative ranges forfor all
if it is closed under relative ranges, finite intersections and unions and contains.
. The triple
is called a labeled space when is accommodating for$A,$$n\geq
1$
, let
For
and
$A$

$\alpha=\alpha_{1}\alpha_{2}\cdots\alpha_{|\alpha|}\in \mathcal{L}^{*}(E)$$\alpha_{i}\cdots\alpha_{j}$$\alpha_{[i_{i}j]}$$\alpha$$\epsilon$$\alpha_{[1,j]}$$\mathcal{B}\subset 2^{E^{0}}$$E^{0}$$A\in \mathcal{B}$$r(A, \alpha)\in \mathcal{B}$$\alpha\in \mathcal{L}^{*}(E)$$\mathcal{B}$$(E, \mathcal{L})$$\mathcal{B}$$(E, \mathcal{L})$$r(\alpha)$$\alpha\in \mathcal{L}^{*}(E)$$(E, \mathcal{L})$$\mathcal{B}$$(E, \mathcal{L}, \mathcal{B})$$B\in 2^{E^{O}}$$AE^{n}=\{\lambda\in E^{n} :s(\lambda)\in A\}, E^{n}B=\{\lambda\in E^{n} :r(\lambda)\in B\}.$which
for $\{v\}E^{n}$ , and will use notations likeandWe writefor $E^{n}\{v\}$ and(receiver
set-finite,
space
meaning.
Alabeledis set-finite
should have their obvious, respectively) is finite.
(
and $l\geq 1$ the set
respectively) if for every
manylabels.
labeled spaceis finite if there are only finitely
weakly left-resolving (left-resolving, respectively) if
We call$AE^{\geq k}$$vE^{n}$$E^{n}v$$vE^{\infty}$$(E, \mathcal{L}, \mathcal{B})$$A\in \mathcal{B}$$\mathcal{L}(AE^{l})$$\mathcal{L}(E^{l}A)$$A$$(E, \mathcal{L}, \mathcal{B})$$(E, \mathcal{L}, \mathcal{B})$$r(A, \alpha)\cap r(B, \alpha)=r(A\cap B, \alpha)$( :is injective for each $v\in E^{0}$ , respectively).
for all $A,$and
Every left-resolving labeled space is weakly left-resolving.
$B\in \mathcal{B}$$\alpha\in \mathcal{L}^{*}(E)$$\mathcal{L}$$r^{-1}(v)arrow \mathcal{A}$Assumptions. We assume that a labeled spacesatisfies the following:(i)is weakly left-resolving.
(ii)
is set-finite and receiver set-finite.
$(E, \mathcal{L}, \mathcal{B})$$(E,\mathcal{L}, \mathcal{B})$$(E, \mathcal{L}, \mathcal{B})$considered in this paper always

15
FINITE SIMPLE$C^{*}$-ALGEBRAS OF LABELED SPACESFor $w\in E^{0}$ , we write $v\sim lw$ if
in [2]. Thendefines anequivalence relation on, and the equivalence classof is called a generalized vertex.
If $k>l,$$v_{m}\in[v]_{t}([2,$is obvious andfor some vertices
Proposition 2.4]).
$\mathcal{L}(E^{\leq l}v)=\mathcal{L}(E^{\leq l}w)ss$$v,$$E^{0}$$[v]_{l}$$[v]_{k}\subset[v]_{l}$$\sim\iota$$v$$[v]_{l}= \bigcup_{i=1}^{m}[v_{\grave{l}}]_{l+1}$$v_{1_{\rangle}}\ldots,$

Notation 2.1. Let
be a labeled graph.
(i) For a labeled space, we denote by
the smallest accommodating set that
contains
and is normal (closed under relative complements). The existence offollows clearly from considering the intersection of all those accommodating sets.
will denote the smallest accommodating set that is closed under relative complements$(E, \mathcal{L})$$\overline{\mathcal{B}}$$(E,\mathcal{L},\mathcal{B})$$\mathcal{B}\cup \mathcal{B}_{sink}$$\overline{\mathcal{B}}$$\overline{\mathcal{E}}$and contains the sets inwill denote the union of all labeled paths
symbol such that $r(\epsilon)=E^{0},$ $r(A, \epsilon)=A$ for all$\{r(\alpha) :(ii)\alpha\in \mathcal{L}^{*}(E)\}.$$\mathcal{L}\#(E)$$\mathcal{L}^{*}(E)$and empty word , where$\epsilon$$\epsilon$

is a$A\subseteq E^{0}.$Proposition 2.2. ([2, Remark 2.1 and Proposition 2.4.(ii)] andbe a labeled graph. Thenis of the form$[8_{:}$

Proposition 2.3]) Let$(E, \mathcal{L})$$A\in\overline{\mathcal{E}}$$A=( \bigcup_{i=1}^{n_{1}}[v_{i}]_{l})\cup(\bigcup_{j=1}^{n_{2}}([u_{j}]_{t})_{sink})\cup(\bigcup_{k=1}^{n_{3}}[w_{k}]\iota\backslash ([w_{k}]_{l})_{sink})$

for some{ source vertices} and. If

has nosinks and sources, contains all generalized vertioes; moreover every
is a finite union of$i\geq 1$ , and $n\geq 1.$
generaiized vertices, that isfor some$v_{i},$$w_{k}\in\Omega_{0}(E):=E^{0}\backslash$$u_{j},$$l\geq 1,$$n_{1},$ $n_{2},$$n_{3}\geq 0$$(E, \mathcal{L})$$\overline{\mathcal{E}}$$A\in\overline{\mathcal{E}}$$v_{i}\in E^{0},$$A= \bigcup_{i=1}^{n}[v_{i}]_{l}$2.2. Labeled graphDefinition 2.3.

Let
projections$(E, \mathcal{L}, \mathcal{B})$$C^{*}$-algebras.

Definition $4.1|, [2,$ Remark $3.2], [3,$ Definition $2.1], and [8,$ Definition $2.4])$
be a labeled space such that
. A representation of
consists ofand partial isometries $\{s_{a} : a\in A\}$ such that for $A,$and$([1_{\}}$$\overline{\mathcal{E}}\subseteq \mathcal{B}$$\{p_{A} :$(E, \mathcal{L}, \mathcal{B})$A\in \mathcal{B}\}$$B\in \mathcal{B}$$a,$ $b\in \mathcal{A},$(i)(ii) $p_{A}s_{a}=s_{a}p_{r(A,a)},$
(iii)
and
(iv) for each$p_{\emptyset}=0,$$p_{\fbox{Error::0x0000}}4p_{B}=p_{A\cap B}$$s_{(X}^{*}s_{C\lambda}=p_{r(a)}$, and $p_{A\cup B}=p_{A}+p_{B}-p_{A\cap B},$$s_{a}^{*}s_{b}=0$

unless $a=b,$$A\in \mathcal{B},$$p_{A}= \sum_{a\epsilon \mathcal{L}(AE^{1})}s_{a}p_{r(A,a)}s_{a}^{*}+p_{A_{sink}}.$By$C^{*}(p_{A}, s_{a})$we denote the$C^{*}$-algebra generated by$\{s_{a},p_{A} :a\in \mathcal{A}, A\in \mathcal{B}\}.$

Remark 2.4, Let
be a labeled space such that(i) There exists a-algebra generated by a universal representationof(see the proof of [1, Theorem 4.5] and [7, Remark 2.5]), If
is a universal
representation of, we call, denoted, the labeled graph-algebra of. Note thatand $p_{A}\neq 0$ for
, and
and
that
if and only if. By definition of representation and[1, Lemma 4.4],$\overline{\mathcal{E}}\subseteq \mathcal{B}.$$(E, \mathcal{L},\mathcal{B})$$C^{*}$$\{s_{a},p_{A}\}$$(E, \mathcal{L}, \mathcal{B})$$\{s_{a},p_{\fbox{Error::0x0000}}4\}$$(E, \mathcal{L}, \mathcal{B})$$C^{*}$$(E, \mathcal{L}, \mathcal{B})$$s_{\alpha}p_{A}s_{\beta}^{*}\neq 0$$C^{*}(s_{a},p_{A})$$s_{a}\neq 0$$C^{*}(E, \mathcal{L}, \mathcal{B})$$a\in \mathcal{A}$$C^{*}(E, \mathcal{L}, \mathcal{B})=\overline{spo.n}\{s_{\alpha}p_{A}s_{\beta}^{*}:\alpha, \beta\in \mathcal{L}^{\#}(E), A\in \mathcal{B}\}$where$s_{\epsilon}$$A\in \mathcal{B},$$A\neq\emptyset$$A\cap r(\alpha)\cap r(\beta)\neq\emptyset$

is regarded as the unit of the multiplier algebra of,$C^{*}(\mathcal{B}, \mathcal{L}, \mathcal{B})$

(1)

.

16
J. A. JEONG, E. J. KANG, S. H. KIM, AND G. H. PARK(ii) Universal property of$\gamma$:$C^{*}(E, \mathcal{L}, \mathcal{B})=C^{*}(s_{a},p_{A})$$\gamma_{z}(s_{a})=zs_{a}$

for

and

$a\in \mathcal{L}(E^{1})$$A\in \mathcal{B}$defines a strongly continuous action

such that$\mathbb{T}arrow Aut(C^{*}(E, \mathcal{L}, \mathcal{B}))$and$\gamma_{z}(p_{A})=p_{\mathcal{A}}$, which we call the gauge action.
is equal to(iii) The fixed point algebra of the gauge action$\gamma$$C^{*}(E, \mathcal{L}, \mathcal{B})^{\gamma}=\overline{span}\{s_{\alpha}p_{A}s_{\beta}^{*}:|\alpha|=|\beta|, A\in \mathcal{B}\}$and it is an AF algebra. Moreover, sinceconditional expectation$\mathbb{T}$(2),

is a compact group, there exists a faithful$\Psi:C^{*}(E, \mathcal{L}, \mathcal{B})arrow C^{*}(E, \mathcal{L}, \mathcal{B})^{\gamma}.$(iv) From Definition 2.3(iv), we have for each $n\geq 1,$$p_{A}= \sum s_{\alpha}p_{r(A,\alpha)}s_{\alpha}^{*}+\sum_{)\alpha\in \mathcal{L}(AE^{n})\gamma\in \mathcal{L}(AE\leq n-1}s_{\gamma}p_{r(A,\gamma)_{sink}}s_{\gamma}^{*},$

where$\sum_{\gamma\in \mathcal{L}(AE^{0})}s_{\gamma}p_{r(A,\gamma)_{\epsilon ink}}s_{\gamma}^{*}:=p_{A_{sink}}.$is agreeable for a, a path
Recall [2, 7] that for a labeled space
and its initial path , and $k\geq 1.$
for some
if
generalized vertex$l\geq 1,$ $v\in E^{0}$ , is disagreeable inis said to be disagreeable if every
labeled spacewhich
the sense that there is an $N\geq 1$ such that for all $n\geq N$ there is a pathis not agreeable.
$\alpha\in \mathcal{L}([v]_{l}E^{\geq 1})$$(E, \mathcal{L},\overline{\mathcal{E}})$$\beta\in \mathcal{L}([v]_{l}E^{\leq l})$$\alpha=\beta^{k}\beta’$$[v]_{l}$$A$$\beta’$$[v]_{l},$$(E, \mathcal{L},\overline{\mathcal{E}})$$\alpha\in \mathcal{L}([v]_{l}E^{\geq n})$such that $p_{A}\neq 0$ foris disagreeable, every representation
Remark 2.5. If
isomorphic togives rise to a -algebra
all non-empty set
is disagreeable if there is no
Theorem 5.5] and , Corollary 2.5]). A labeled space$([8,$ Proposition $4.12])$ .
repeatable paths in$\{s_{a_{1}}p_{A}\}$$(E, \mathcal{L},\overline{\mathcal{E}})$$C^{*}(E, \mathcal{L},\overline{\mathcal{E}})([2,$$C^{*}(s_{a},p_{A})$$C^{*}$$A\in\overline{\mathcal{E}}$$(E, \mathcal{L}, \overline{\mathcal{E}})$$[^{?}$$(E, \mathcal{L})$2.3. $K$-theory of labeled graph C’-algebras. $K$-theory of labeled graph -algebras was
be a normal labeled space. Then the set
obtained in [3]. Let have no sinks and:4.4] the linear map $(1-\Phi)$ :[3,
by,Theoremandgiven in (2) of [3] coincides with$C^{*}$$E$$\mathcal{B}_{J}$$(E, \mathcal{L}, \mathcal{B})$$span_{Z}\{\chi_{A}$$\mathcal{B}$$A\in \mathcal{B}\}arrow span_{Z}\{XA : A\in \mathcal{B}\}$

given by$(1- \Phi)(\chi_{A})=\chi_{A}-\sum_{a\in \mathcal{L}(AE^{1})}\chi_{r\langle A,a)}, A\in \mathcal{B}$determines the $K$-groups of$C^{*}(E, \mathcal{L}, \mathcal{B})$(3)

as follows:$K_{0}(C^{*}(E, \mathcal{L}, \mathcal{B}))\cong span_{\mathbb{Z}}\{\chi_{A} :$K_{1}(C^{*}(E, \mathcal{L}, \mathcal{B}))\underline{\simeq}ker(1-\Phi)$

A\in \mathcal{B}\}/{\rm Im}(1-\Phi)$

.

(4)
(5)

In (4), the isomorphism is given by $[p_{A}]_{0}\mapsto\chi_{A}+{\rm Im}(1-\Phi)$ for$A\in \mathcal{B}.$2.4. Generalized Morse sequences. We review from [10] definitions and basic properties of(generalized) Morse sequences. Let$\Omega:=\{\omega=\cdots\omega_{-2}\omega_{-1}\omega_{0}\omega_{1}\cdots:\omega_{i}\in\{0, 1\}, i\in \mathbb{Z}\}$be the space of all two-sided sequences of zeros and ones, and let$\Omega_{+}:=\{x=x_{0}x_{1}\cdots :x_{i}\in\{0, 1\}, i\geq 0\}$denotes the set of all finite blocks (finite sequences) of zeros
the space of onesided sequences.
$(x\in\Omega+,$ respectively) , the$:=n+1$ . For, its length isand ones. For$\mathfrak{B}$$b=b_{0}\cdots b_{n}\in \mathfrak{B}$$|b|$$\omega\in\Omega$

17
FINITE SIMPLE$C^{*}$-ALGEBRAS OF LABELED SPACES

set of all finite blocks appearing in $(v(x,$ respectively) will be denoted byrespectively) .
, the set of all two-sided sequences such thatFor
is denoted by$\mathfrak{B}_{\omega}(\mathfrak{B}_{x},$$x\in\Omega_{+}$$\mathfrak{B}_{\omega}\subseteq \mathfrak{B}_{x}$$\omega$$\theta_{x}:=\{\omega\in\Omega:\mathfrak{B}_{\omega}\subseteq \mathfrak{B}_{x}\}.$, we writeForof . Similarly,$\omega\in\Omega$which is a block at the position
mean the infinite sequencesand$\omega_{[t_{1},t_{2}]}:=\langle JJt_{1}\ldots\omega_{t_{2}}\in \mathfrak{B}_{\omega}$$\omega$$\omega_{[t_{1},\infty)}$and$t_{1}(t_{1}\leq t_{2})$$\omega_{t_{1}}\omega_{t_{1}+1}\cdots$$\omega_{\langle-\infty,t_{2}]}$$\omega_{t_{2}-1}\omega_{t_{2}},$

respectively.
The space (and similarly) endowed with the product topology becomes a totally disconnected compact Hausdorﬀ space such that the clopen cylinder sets$\Omega$$\Omega_{+}$$t[b]:=\{\omega\in\Omega:\omega_{[t,t+n]}=b\},$$t\in \mathbb{Z},$$b\in \mathfrak{B},$$|b|=n+1\geq 1$ , form a base for the topology. For convenience, we use thefollowing notation:$[.b]:=0[b], [b.]:=-|b|[b], [b.c]:=-|b|[bc]$. Note that on the right side of the dot is the zeroth position.
for
The shift. map
given by, is easily seen to be a homeomorphism. For, the closure of the orbit ofwill be denoted by
Each block
defines a block , the mirror image of , such that(mod2). For, tbe }) roduct bof and denotes the block (of length) obtainedby putting $n+1$ copies of either or next to each other according to the rule of choosing theith copy as if $c_{9}=0$ and if $c_{i}=1.$
Let
be a sequence of b:ocks with length
such thatfor all $i\geq 0$ . Then one can consider (one-sided) recurrent sequence of the form$b,$$c\in \mathfrak{B}$$T:\Omegaarrow\Omega$$\omega\in\Omega,$$(T\omega)_{i}=\omega_{i+1},$$i\in \mathbb{Z}$$\omega\in\Omega$$\theta_{\omega}$$:=\{T^{i}(\omega) :$\omega$i\in \mathbb{Z}\}\subseteq\Omega.$$\tilde{b}$$b\in \mathfrak{B}$$\tilde{b}_{i}=b_{i}+1$$b$$c=c_{0}\cdots c_{n}\in \mathfrak{B}$$b$$\cross c$$|b|\cross|c|$$c$$\tilde{b}$$b$$\tilde{b}$$b$$|b^{i}|\geq 2$$\{b^{t} :=b_{0}^{i}\cdots b_{|b^{l}|-1}^{i}\}_{\^{i}\geq 1}c;\mathfrak{B}$$b_{0}^{i}=0$$a$$x=b^{0}\cross b^{1}\cross b^{2}\cross\cdots\in\Omega+$$(see [10,$ Definition $7 We$ call such $an x=b^{0}\cross b^{1}\cross b^{2}\cross\cdots\in\Omega+a ($generalized) one-sided

Morse sequence if it is non periodic and, whereis the relative$(see [10, 1).338])$ .
frequency of occurrence of $(a=0$ or 1 in
Recall thatis uniquely ergodic if
admits exactly one -invariant probability measure. Such a unique measure is automatically ergodic.
$\sum_{i=0}^{\infty}\min(r_{0}(b^{i}), r_{1}(b^{\grave{l}}))=\infty$$a$$ﬀ_{\omega}$$r_{tl}(b)$$b\in \mathfrak{B}$

$)$

$T$

$ﬀ_{\omega}$$m_{\omega}$

Theorem 2.6. ([10, Lemma 2, Lemma 4, Theorem 3]) Let
be a non-periodic recurrent
sequence. Then we have the following:(i)is almost periodic; for any cytinder set, there exists $d\geq 1$ such that for
any $n\geq 0,$ $T^{n+j}x\in[.b]$ for some $0\leq j\leq d.$(ii) There exists
with. Moreover, \’is one-sided Morse sequence if andonly if
minimal and uniquely ergodic, and \’if this is the case, then$x\in\Omega_{+}$$x$$[.b]_{J}b\in \mathfrak{B}_{x}$$\omega\in\beta_{x}$$x$$x=\omega_{[0,\infty\rangle}$$a$$ﬀ_{\omega}\dot{?}s$$\beta_{\omega}=g_{x}.$Definition 2.7. By a generalized Morse sequence, we mean a two-sided sequencethatis a one-sided Morse sequence and$\omega\in\Omega$such$\mathfrak{B}_{\omega}=\mathfrak{B}_{x}.$$x:=\omega_{[0,\infty)}$

Remark 2.8. For a generalized Morse sequence , the unital commutative AF algebraof all continuous functions on
admits (tracial) state$\omega$$0_{\omega}$$a$$f \mapsto\int_{l}p_{\omega}fdrr\iota_{\omega}:C(\theta_{\omega})arrow \mathbb{C}$$C(O_{\omega})$

18
J. A. JEONG, E. J. KANG, S. H. $KlM$ , AND G. H. PARK

. Sincewhich we also write, and hence$m_{\omega}$$m_{\omega}$

is -invariant, it easily follows that$T$$m_{\omega}(\chi_{db]})=m_{\omega}(\chi_{t[b]}\circ T)=$$m_{\omega}(\chi_{t+1[b]})$(6)$m_{\omega}(\chi_{\iota[b]})=m_{\omega}(\chi_{[.b]})$holds for all$t\in \mathbb{Z}$and$b\in \mathfrak{B}_{w}.$

Example 2.9. (Thue-Morse sequence) Let
sequence$b^{i}$$:=01\in \mathfrak{B}$for all $i\geq 0$ . Then the recurrent$x:=b^{0}\cross b^{1}\cross b^{2}\cross\cdots=01\cross b^{1}\cross\cdots=0110\cross b^{2}\cross\cdots=01101001\cross b^{3}\cross\cdots$

is a one-sided Morse sequence called the Thue-Morse sequence and$\omega :=x^{-1}.x=\cdots 10010110.011010011001\cdots\in\beta_{x}$$:=\cdots x_{2}x_{1}x_{0}$ is the sequence obtained by writing $x=$
is a generalized Morse sequence, where
in reverse order. In fact, is the sequence constructed from the proof of Theorem2.6(ii)for any
has no blocks of the form
(see [10, Lemma 4 and it is well known [6] that$x^{-1}$$\omega$$x_{0}x_{1}\cdots$$bbb_{0}$$\omega$$b=b_{0}\cdots b_{|b|-1}\in \mathfrak{B}_{\omega}.$

3. AF LABELED GRAPH$C^{*}$-ALGEBRAS

in a directed graph $E$ is a loop if $s(x)=r(x)$ . It is well known
Recall that a path
algebra C $(E)$ to be AF it is a suﬃcient and necessary[14, Theorem 2.4] that for a graphcondition that $E$ has no loops. To find conditions of a labeled space which arises an AF
algebras, we define following generalized notion of loop.
$x\in E^{\geq 1}$$C^{*}arrow$$C^{*}-$

a labeled path.
be a labeled space andDefinition 3.1. Let.
if(a)is a generatized loop at.
if it is a generalized loop such that(b) is a loop at
holds:
following
if
one
of
the
exit
(c) A loop at
has an,(i)(ii)
for some $i=1$ , . . . ,.
(iii)$\alpha\in \mathcal{L}^{*}(E)$$(E, \mathcal{L}, \mathcal{B})$$A\in \mathcal{B}$$\alpha$$\alpha\in \mathcal{L}(AE^{\geq 1}A)$$A\subset r(A, \alpha)$$A\in \mathcal{B}$$\alpha$$A\in \mathcal{B}$$\alpha$$\{\alpha_{[1,k]}:1\leq k\leq|\alpha|\}\subsetneq \mathcal{L}(AE^{\leq|\alpha|})$$|\alpha|,$$r(A, \alpha_{[1_{\}}x]})_{sink}\neq\emptyset$$A\subsetneq r(A, \alpha)$

be a representation of
Remark 3.2. Let(i) A generalized loop at a minimal set
might have (generalized) loop
loops at all.
then(ii) If is a loop at$(s_{a},p_{A})$$\alpha$$(E, \mathcal{L})$$a$$A\in \mathcal{B}$$\alpha$.
is necessarily a loop. A labeled grapheven when the underlying graph $E$ has no$(E, \mathcal{L}, \mathcal{B})$$A\in \mathcal{B}$$\alpha$$p_{A}\leq p_{r(A,\alpha)}.$Proposition 3.3. Let
be a labeled graph and.
is an infinite projection in$(E, \mathcal{L})$$\alpha$be a loop at$A\in\overline{\mathcal{E}}$with an$ex\iota t$. Then$C^{*}(E, \mathcal{L},\overline{\mathcal{E}})$$p_{r(A,\alpha)}$Theorem 3.4. If$C^{*}(E, \mathcal{L},\overline{\mathcal{E}})$is an $AF$ algebra, the labeled space$(E, \mathcal{L},\overline{\mathcal{E}})$

has no loops.

contains
with the trivial labeling
Since the accommodating set of a labled graph$v\in E^{0}$ ,the following are equivalent for a path $x=x_{1}\cdots x_{m}\in$
all the single vertex sets$\overline{\mathcal{E}}$$\{v\},$:
is a loop in $E,$

$E^{\geq 1}(=\mathcal{L}_{id}^{*}(E))$(i)(ii) $\{r(x)\}=r(\{r(x)\}, x)$ ,$x$$(E, \mathcal{L}_{id})$$\mathcal{L}_{id}$19
FINITE SIMPLE

(iii)
(iv)

$x$

is repeatable, that is,$C^{*}$-ALGEBRAS OF LABELED SPACES$x^{n}\in E^{\geq 1}$

for all $n\geq 1,$$(A_{1}x_{1}A_{2}x_{2}\cdots A_{m}x_{m})^{n}(A_{1}x_{1}A_{2}x_{2}\cdots A_{i}x_{i})\in \mathcal{L}_{id}^{*}(E)$for ali $n\geq 1$ and$1\leq i\leq m_{\}}$

where$A_{i}=\{s(x_{i})\}\in\overline{\mathcal{E}}.$

From this we can obtain several equivalent conditions for a graph $C^{*}-alge$ })$raC^{*}(E)$ to be AF
as follows.

so that
Proposition 3.5. Let
be a labeted space with the trivial labeling$C^{*}(E)$ . Then the foltowing are equivalent.
(i)is $AF,$(ii) $E$ has no loops,,(iii) $A\not\subset r(A, x)$ for alland,(iv) there are no repeatable paths inand $K\geq 1$ , there is an $m_{0}\geq 1$(v) ifa finite collection of sets from$attn>m_{0}.$such that
for$(E, \mathcal{L}_{td},\overline{\mathcal{E}})$$C^{*}(E, \mathcal{L}_{id},\overline{\mathcal{E}})\cong$$\mathcal{L}_{id}$$C^{*}(E, \mathcal{L}_{id},\overline{\mathcal{E}})$$A\in\overline{\mathcal{E}}$$x\in \mathcal{L}_{id}^{*}(E)$$\mathcal{L}_{\dot{t}d}^{*}(E)$$\{A_{1}, .

.

$\overline{\mathcal{E}^{0}}$., A_{m}\}?\dot{s}$$A_{i_{1}}E^{\leq K}A_{i_{2}}\cdots E^{\leq K}A_{\grave{\iota}_{n+1}}=\emptyset$Let$A_{1}E^{\geq 1}A_{2}\cdots E^{\geq 1}A_{n+1}$

denote the following set$\{x=x_{1}x_{2}\cdots x_{n}\in E^{\geq 1}:x_{k}\in A_{k}E^{\geq 1}A_{k+1}, 1\leqk\leq n\}.$Theorem 3.6. Letin such that$(E, \mathcal{L})$be a labeled graph. Assume that if$A_{1},$ $A_{2}$, . . . as a sequence of sets$\overline{\mathcal{E}}$$A_{1}E^{\geq 1}A_{2}E^{\geq 1}A_{3}\cdots E^{\geq 1}A_{n}\neq\emptyset$

for all$n\geq 1$, the set$\{A_{1}, A_{2}, .

.

.

\}$

For a labeled $gl\cdot aphC^{*}$ -algebra
the ideal of$C^{*}(E, \mathcal{L},\overline{\mathcal{E}})$

Lemma 3.7. Let
sinks or sources. For $A,$
many pathsin

is infinite. Then

$C^{*}(E, \mathcal{L},\overline{\mathcal{E}})=C^{*}(s_{a\rangle}p_{A})$generated by the projection$B\in\overline{\mathcal{E}}$, we havesuch that$p_{A}\in I_{B}$$p_{A}$

and a set

Recall that the set
the setprojection:$p[v]\ell_{\epsilon,nk}$

be the$C^{*}(E, \mathcal{L},\overline{\mathcal{E}})=C^{*}(s_{f1},p_{A})$$\alpha^{n}\in \mathcal{L}^{*}(E)$$p_{r(\alpha^{m})\backslash r(\alpha^{m+1})}$$\{A\in\overline{\mathcal{E}}$for$atln\geq 1$. If$C^{*}$$[v]\iota_{si\ddagger*}$of all sinks of$A\subseteq[v]_{t_{sink}}\}$. The ideal$[v]_{i}$$C^{*}$$p_{r(\alpha^{m})}$for some $m\geq 1$ , then$(E, \mathcal{L})$.

and let-algebra of a labeled graphby
ideal
generatedbelongtothedoes not$AF,$
is not$(E, \mathcal{L})$$C^{*}(E, \mathcal{L},\overline{\mathcal{E}})$is a member ofof$I_{[v]_{l_{8}tnk}}$$\overline{\mathcal{E}}$and thatdenotes
generated by the$\overline{\mathcal{E}}\cap[v]_{t_{StY1}k}$$C^{*}(E, \mathcal{L},\overline{\mathcal{E}})=C^{*}(p_{A}, s_{a})$is equal to$\overline{span}\{s_{\alpha}p_{A}s_{\beta}^{*}:\alpha,$$\beta\in \mathcal{L}^{\#}(E)$and$A\in\overline{\mathcal{E}}\cap[v]\iota_{sink}\}.$

Lemma 3.9. Let
be a labeled space and $v\in\Omega_{0}(E)$ . If
$i=1_{\}}\ldots$ , $N,$
finitely many min\’imal sets$(B, \mathcal{L},\overline{\mathcal{E}})$$[v]\iota_{sink}$

is the disjoint union of$A_{i}\in\overline{\mathcal{E}},$$I_{[v]_{i_{sink}}}=\oplus_{i=1}^{N}\overline{spar\}}\{\mathcal{S}_{\alpha}p_{A}.s_{\beta}^{*}:

where$\mathcal{L}(ffiA_{i})$$I_{A}$

as before.

$\bigcup_{|\beta|=N}r(A, \beta)c\bigcup_{i=1}^{n}r(B, \mu_{i})$$\alpha\in \mathcal{L}^{*}(E)$, we denote by$\mathcal{L}(BE^{\geq 0})$$\{\mu_{i}\}_{i=1}^{n}$Lemma 3.8. Let
satisfy
a projection$A\in\overline{\mathcal{E}}$with no-algebra of a labeled graphif and only if there exist an $N\geq 1$ and finitely

be the$C^{*}(E, \mathcal{L},\overline{\mathcal{E}})=C^{*}(s_{a},p_{A})$is $AF.$$C^{*}(E, \mathcal{L}_{\}}\overline{\mathcal{E}})$$:=\{\epsilon\}.$\alpha, \beta\in \mathcal{L}^{\#}(E)\}\cong\oplus_{i=1}^{N}\mathcal{K}(l^{2}(\mathcal{L}(E^{\geq 0}A_{i})))$,

20
J. A. JEONG, E. J. KANG, S. H. KIM, AND G. H. PARK

Proposition 3.10. Let
be a finite labeled space such that there exists an $l\geq 1$ for
which. Thenhas no generalized loops at
for all$(E, \mathcal{L},\overline{\mathcal{E}})$$[v]_{l}\in\Omega_{1}(E)$$[v]_{l}$$(E, \mathcal{L}, \overline{\mathcal{E}})$$C^{*}(E, \mathcal{L}, \overline{\mathcal{E}})\cong\oplus_{[v]_{l}\in\Omega_{l}(E)}I_{[v]\iota_{\epsilon ink}}.$Moreover, the ideal$I_{[v]_{l_{sink}}}$is finite dimensional whenever$\overline{\mathcal{E}}\cap[v]\iota_{sink}$

4. $NoN-AF$ FINITE SIMPLE LABELED GRAPH$C^{*}$

is a finite set.

-ALGEBRAS-algebra is said to be infinite if it has an infinite projection. A unital-algebra
is purely infinite if for each nonzero positive element $a\in A$ there is a $b\in A$ satisfying$b^{*}ab=1$ . A purely infinite-algebra
is always simple since the ideal generated by any
positive nonzero element contains the unit of A. (For nonsimple purely infinite-algebras see[11, 12 It is an easy observation that a simple unital -algebra is purely infinite if and only
if every nonzero hereditary -subalgebraof has a projection
equivalent
to the unit $1=(a^{1/2}b)^{*}(a^{1/2}b)$ . Thus if is purely infinite, every nonzero projection is always
infinite. A simple C’-algebra without unit is called purely infinite if every nonzero hereditary-subalgebra contains an infinite projection.
We call a -algebra finite when has no infinite projections. A simple unital -algebrawith a tracial state ( is automatically faithful since is simple) is always finite because
the faithfulness of implies that if a projection $p\in A$ is equivalent to its subprojection $q\leq p$
in , with $p=vv^{*}$ and $q=v^{*}v$ for $v\in A$ , then $\tau(p-q)=\tau(vv^{*}-v^{*}v)=0$ and so $p-q=0$
by faithfulness of
Besides commutative -algebras, all finite dimensional -algebras are obviously finite, andmoreover all AF algebras are also finite. On the other hand, the Cuntz-algebras, 3, .. . ,or more generally simple Cuntz-Krieger algebras are well known to be purely
infinite.
In [2, Proposition 7.2], Bates and Pask provide an example of a simple unital purely infinite
labeled graph -algebra which is not isomorphic to any unital graph -algebra. We also know
from [16] that there exist simple higher rank graph -algebras which are neither AF nor purely
infinite; there exist such simple -algebras which are stably isomorphic to irrational rotation
algebras or Bunce-Deddens algebras. This fact leads us to ask if there exists a simple unitallabeled graph -algebra which is neither AF nor purely infinite. To this question we answer inTheorem 4.4 that there really exists a simple unital finite, but non-AF labeled graph -algebra. This is a -algebra associated to a labeled space
which is labeled
by a generalized Morse sequence
Throughout this section,
will denote the following graph:

Recall$C^{*}$$C^{*}$$A(\neq \mathbb{C})$$A$$C^{*}$$C^{*}$$A$$C^{*}$$\overline{aAa}$$C^{*}$$A$$a^{1/2}b(a^{1/2}b)^{*}$$A$$C^{*}$$A$$A$$C^{*}$$A$$\tau$$C^{*}$$A$$\tau$$\tau$$A$$\tau.$$C^{*}$$C^{*}$$\mathcal{O}_{n}(n=$$2$$\infty)[4]$$C^{*}$$C^{*}$$C^{*}$$C^{*}$$C^{*}$$C^{*}$$C^{*}$$C^{*}(E_{Z}, \mathcal{L}_{\omega}, \overline{\mathcal{E}}_{\mathbb{Z}})$$(E_{\mathbb{Z}}, \mathcal{L}_{\omega},\overline{\mathcal{E}}_{Z})$$\omega.$$E_{Z}$. ..

$\bulletarrow^{-4}\bulletarrow^{-3}\bulletarrow^{-2}\bulletarrow^{-1}\bulletarrow^{0}\bulletarrow^{1}\bulletarrow^{2}\bulletarrow^{3}$$\bullet\cdots$$v_{-4} v_{-3} v_{-2} v_{-1} v_{0} v_{1} v_{2} v_{3} v_{4}$Given a two-sided sequencegraph$(E_{Z}, \mathcal{L}_{\omega})$$\omega=\cdots\omega_{-1}\omega_{0}\omega_{1}\cdots\in\Omega$.

of zeros and ones, we obtain a labeledshown below$(E_{Z}, \mathcal{L}_{\omega})\cdots\bullet\underline{\omega_{-4}}.arrow\omega_{-3}.$$\underline{\omega_{-2}}.$$arrow\bullet\omega_{-1}arrow^{\omega_{0}}\bulletarrow^{\omega_{1}}\bulletarrow^{\omega_{2}}\bulletarrow^{\omega_{3}}$$\bullet$. .. ,$v_{-4} v_{-3} v_{-2} v_{-1} v_{0} v_{1} v_{2} v_{3} v_{4}$where the labeling map
have a labeled space
relative complements.

$\mathcal{L}_{\omega}$:, 1 is given by
forwith the smallest accommodating set$E_{Z}^{1}arrow\{0$$(E_{\mathbb{Z}}, \mathcal{L}_{\omega},\overline{\mathcal{E}}_{Z})$$\}$$\mathcal{L}_{\omega}(n)=\omega_{n}$. Then we akowhich is closed under$n\in E_{Z}^{1}$$\overline{\mathcal{E}}_{Z}$21
FINITE SIMPLE$C^{*}$-ALGEBRAS OF LABELED SPACES

be the labeled graph -algebra associated with the labeledLet
generalizedaspaceMorse sequence . Then by (2) the fixed point algebra of
and
gauge
generated by elements of
forxn
the
action$cx=\beta$, and hence
which is nonzero only when$C^{*}$$C^{*}(E_{Z}, \mathcal{L}_{\omega},\overline{\mathcal{E}}_{\mathbb{Z}})=C^{*}(s_{a},p_{A})$$(E_{\mathbb{Z}},\mathcal{L}_{c)}, \overline{\mathcal{E}}_{\mathbb{Z}})0\xi$$\omega$$t\}_{1}e$$\gamma\dot{i}S$$C^{*}(\ ,\mathcal{L}_{\omega},\overline{\mathcal{E}}_{Z})^{\gamma}=\overline{span}\{s_{\alpha}p_{\mathcal{A}}s_{\alpha}^{*} :Moreover$C^{*}(E_{\mathbb{Z}}, \mathcal{L}_{\omega},\overline{\mathcal{E}}_{\mathbb{Z}})^{\gamma}$$A\subset r(\alpha)\cap r(\beta))$$s_{a}p_{A}s_{\beta}^{*}(|\alpha|=|\beta|$

A\in\overline{\mathcal{E}}_{Z}, Acr(\alpha)\}.$

is easily seen to be a commutative C’-algebra. For each $k\geq 1$ , let$F_{k}:=span\{s_{\alpha}p_{r(\alpha’\alpha)}s_{\alpha}^{*}:a, \alpha’\in \mathcal{L}_{\omega}(E_{Z}^{k})\}.$

inare linearly independent and actually orthogonal
The (finitely many) elements. Moreoverisis a finite dimensional subalgebra of
to each other so that
becausea subalgebra of$F_{k}$$s_{\alpha}p_{r(\alpha\alpha\rangle}s_{\alpha}^{*}$$F_{k}$$C^{*}(E_{Z;}\mathcal{L}_{\omega},\overline{\mathcal{E}}_{Z})^{\gamma}$$F_{k}$$F_{k+1}$$s_{\alpha}p_{r(\alpha’cx)}s_{\alpha}^{*}= \sum_{b\in\{0,1\}}s_{\alpha b}p_{r(\alpha’\alpha b)}s_{\alpha b}^{*}=\sum_{(x,b\in\{0,1\}}s_{\alpha b}p_{r(a\alpha’\alpha b)}s_{\alpha b}^{*}.$. .. of finite dimensional C’-algebras,
This gives rise to an inductive sequence
where the connecting maps : $F_{k}arrow F_{k+1}$ are inclusions for $k\geq 1$ , from which we obtain an. ThenAF algebra$F_{1}arrow^{1}\iota F_{2}arrow t2$$b_{k}$$\underline{1in3}^{F_{k}}\prime$$C^{*}(E_{Z},\mathcal{L}_{\omega},\overline{\mathcal{E}}_{Z})_{\frac{1ix\mathfrak{p}}{\prime}}^{\gamma_{=}}F_{k)}$and thus the fixed point algebra is an AF algebra.

Proposition 4.1. Let
there is a surjective isomorphism$(E_{Z}, \mathcal{L}_{\omega},\overline{\mathcal{E}}_{Z})$be the labeled space of a generalized Morse sequence$\omega$(7)$p:C^{*}(E_{\mathbb{Z})}\mathcal{L}_{\omega}, \overline{\mathcal{E}}_{\mathbb{Z}})^{\gamma}arrow C(\beta_{\omega})$such that$p(s_{\alpha}p_{r(\alpha’\alpha\rangle}s_{\alpha}^{*})=\chi[\alpha^{t}.\alpha]$

Lemma 4.2. Letfor$s_{\alpha}p_{r(\alpha’\alpha)}s_{\alpha}^{*}\in F_{k},$$k\geq 1.$

be the labeled space of a generalized Morse sequence and let
be the isomorphism in (7). Then the unique -invariant ergodic
defines a tracial state$(E_{Z}, \mathcal{L}_{\omega},\overline{\mathcal{E}}_{Z})$$\omega$$T$$p:C^{*}(E_{Z}, \mathcal{L}_{\omega},\overline{\mathcal{E}}_{\mathbb{Z}})^{\gamma}arrow C(P_{\omega})$measure$m_{\omega}$:$C(\rho_{\omega})arrow \mathbb{C}$. Then$\tau_{0}:=m_{\omega}o\rho:C^{*}(E_{\mathbb{Z}}, \mathcal{L}_{\omega\rangle}\overline{\mathcal{E}}_{Z})^{\gamma}arrow \mathbb{C}$on the fixed point algebra$C^{*}(E_{Z}, \mathcal{L}_{\omega}, \overline{\mathcal{E}}_{Z})^{\gamma}$

such that for$\alpha,$$\beta\in \mathcal{L}_{\omega}^{*}(E_{Z})$$\tau_{0}(s_{\alpha}p_{r(\beta\alpha)}s_{\alpha}^{*})=\tau_{0}(p_{r(\beta\alpha)})$,.

The following lemma can be proved by straightforward computation.
Lemma 4.3. Let$(E_{Z},\mathcal{L}_{\omega},\overline{\mathcal{E}}_{Z})$be the labeled space of a generalized Morse sequence$w$. Then$\tau_{0}\circ\Psi:C^{*}(E_{Z}, \mathcal{L}_{\omega},\overline{\mathcal{E}}_{Z})arrow \mathbb{C}$is a tracial state.

Theorem 4.4. Let be a generalized Morse sequence of zeros and ones. Then theis(i) simple unital,(ii) non $AF,$(iii) finite with a unique tracial state which satisfies$\omega$$C^{*}(E_{Z}, \mathcal{L}_{\omega},\overline{\mathcal{E}}_{\mathbb{Z}})$$\tau$$\tau(s_{\alpha}p_{r(\sigma\alpha)}s_{\beta}^{*})=\tau(\Psi(s_{\alpha}p_{r(\sigma\alpha)}s_{\beta}^{*}))=\delta_{\alpha,\beta}\tau(p_{r\langle\sigma\alpha\rangle})$for$\alpha,$$\beta,$$\sigma\in \mathcal{L}_{\omega}^{*}(E_{Z})$.

$C^{*}$-algebra

22
J. A. JEONG, E. J. KANG, S. H. KIM, AND G. H. PARK

In particular,$C^{*}(E_{Z}, \mathcal{L}_{\omega},\overline{\mathcal{E}}_{\mathbb{Z}})$

is not stably isomorphic to a graph$C^{*}$-algebra.

Let
be a generalized Morse sequence. Then the shift map :automorphism:, $\sigma\tau(f)=foT^{-1}$ . In particular, for each$\omega\in\Omega$$T$$\sigma_{T}$$\theta_{\omega}arrow\rho_{\omega}$$c(a_{\omega})arrow C(\rho_{\omega})$

induces an
we have$A\in\overline{\mathcal{E}}_{Z}$$\sigma_{T}(\chi_{A})=\chi_{A}oT^{-1}=\chi_{T(A)}.$The following can be shown by universal property of the labeled graph -algebra
since one can find a representation ofin the crossed product. The
proof will be contained in the revised version of [9]. Note thatis a Cantor system, sothat we can apply the results known in [5] to identify the isomorphism classes of the crossed
products.
$C^{*}$$(E_{Z}, \mathcal{L}_{\omega},\overline{\mathcal{E}}_{Z})$$C^{*}(E_{Z}, \mathcal{L}_{\omega},\overline{\mathcal{E}}_{Z})$$C(\theta_{\omega})X_{\sigma_{T}}\mathbb{Z}$$(\theta_{\omega}, T)$

Theorem 4.5. Let
be a generalized Morse sequence and
There exists an isomorphism$\omega\in\Omega$$T:a_{\omega}arrow a_{\omega}$be the shift map.

$\pi:C^{*}(E_{Z}, \mathcal{L}_{\omega},\overline{\mathcal{E}}_{Z})arrow C(\theta_{\omega})\aleph_{\sigma_{T}}\mathbb{Z}.$REFERENCES[1] T. Bates and D. Pask,[2] T. Bates and D. Pask,

249-274.

$C^{\cdot}$-algebras of labeled graphs, J. Operator Theory, 57(2007), 101-120.
-algebras of labeled graphs II -simplicity results, Math. Scand. 104(2009), no. 2,$C^{*}$[3] T. Bates, T. M. Carlsen, and D. Pask,-algebras of labeied graphs III -K-theory computations,
arXiv: 1203. 3072v2 [math. OA].
[4] J. Cuntz, $K$-theory for certain-algebras, Ann. Math. 113(1981), no. 1, 181-197.
[5] T. Giordano, I. F. Putnam, and C. F. Skau, Topological orbit equivalence and $C*$ -crossed products, J. reine.
angew. Math., 469 (1995), 51-111.
[6] W. H. Gottschalk and G. A. Hedlund, A $charactemzat_{t}on$ of the Morse minimal set, Proc. Amer. Math. Soc.,
15 (1964), 70-74.
[7] J. A Jeong and S. H. Kim, On simple labeled graph-algebras, J. Math. Anal. Appl. 386(2012), 631-640.
[8] J. A Jeong, E. J. Kang and S. H. Kim, AF labeled graph-algebras, J. Funct. Anal. 266(2014), 2153-2173.
[9] J. A Jeong, E. J. Kang, S. H. Kim, and G. H. Park, Finite simple unital non-AF labeled graph-algebras ofgeneralized Morse sequences, arXiv:1504.0345S.
[10] M. Keane, Generalized Morse sequences, Z. Wahrscheinlichkeitstheorie verw. Geb. 10(1968), 335-353.
[11] E. Kirchberg, M. Rrdam, Non-simple purely infinite-algebras, Amer. J. Math. 122(2000), 637-666.
[12] E. Kirchberg, M. Rrdam, Infinite Non-simple-algebras: bsorbing the Cuntz Algebra, Adv. in Math.
167(2002), 195-264.
[13] A. Kumjian and D. Pask, Higher rank graph-algebras, New York J. Math. 6(2000), 1-20.
[14] A. Kumjian, D. Pask and I. Raeburn, Cuntz Kreeger algebras of directed graphs, Pacific J. Math. 184(1998),$C^{*}$$C^{\cdot}$$C^{*}$$C^{*}$$C^{*}$$C^{\cdot}$$C^{*}$$\mathcal{A}$$\mathcal{O}_{\infty}$$C^{*}$

161-174.

[15] A. Kumjian, D. Pask, I. Raeburn and J. Renault Graphs, groupoids, and Cuntz-Krieger algebras, J. Iinnct.
Anal. 144(1997), 505-541.
[16] D. Pask, I. Raeburn, M. Rrdam, and A. Sims, Rank-two graphs whose-algebras are direct limits of circle
algebras, J. Funct. Anal. 239(2006), 137-178.
$C^{*}$

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