数理解析研究所講究録 1291 巻 2002 年 73-83

73

Continuous graphs and crossed products ofCuntz algebras.
Takeshi Katsura ( 勝良 健史)
Department of Mathematical Sciences
University of Tokyo, Komaba, Tokyo, 153-8914, JAPAN-mail:kat su@ms.u-tokyo.ac.jp$\mathrm{e}$0Introduction
In [Kal, Ka2, Ka3], the author examined the structure of crossed products of Cuntzalgebras by s0-called quasi-free actions of abelian groups. Recently, he introduced anew
class of ’-algebras which are arising from continuous graphs [Ka4]. These (7’-algebras
are generalization of graph algebras [KPRR, KPR, FLR] and homeomorphism (7’-algebras[T3, T4]. The above crossed products are examples of -algebras arising from continuous
graphs. From this point of view, some of results in [Kal] and [Ka3] can be considered asacontinuous counterpart of ones in [BHRS] and [HS]. This observation is further studied
in [Ka5] for more general settings.
In this short article, we give adefinition of continuous graphs and C’-algebras associated with them, and then discuss how the results in [Kal] and [Ka3] can be interpreted
in terms of continuous graphs.
$C$$C^{*}$

1C’-algebras arising from continuous graphs
be locally compact (Hausdorff) spaces. Amap $d:E^{1}arrow$
andDefinition 1.1 Let
is said to be locally homeomorphic if for any $e\in E^{1}$ , there exists aneighborhood ofsuch that the restriction of on is ahomeomorphism onto $d(U)$ and that $d(U)$ is aneighborhood of ci(e).
$E^{1}$$E^{0}$$U$$E^{0}$

$d$

$e$

$U$

Every local homeomorphisms are continuous and open.

Definition 1.2 ([Ka4, Definition 2.1]) Acontinuous graph $E=(E^{0}, E^{1}, d, r)$ consists: $E^{1}arrow E^{0},$ and, alocal homeomorphismof two locally compact spaces
continuous map $r:E^{1}arrow E^{0}$ .
$E^{0},$ $E^{1}$$d$$\mathrm{a}$are not necessarily surjective nor injective. We think that:
Note thatis aset of edges and that an edge $e\in E^{1}$ is directed from its
is aset of vertices anddomain $d(e)\in E^{0}$ to its range $r(e)\in E^{0}$ . From ahomeomorphism on alocally compact$d,$ $r$$E^{0}$$E^{1}arrow E^{0}$$E^{1}$$\sigma$

74
space $X$ , we can define acontinuous graph $E=(E^{0}, E^{1}, d, r)$ by $E^{0}=E^{1}=X,$
and $r=\sigma$ . In this sense, acontinuous graph can be considered as ageneralization of
dynamical systems.
such thatthe set of continuous functions 4of
Let us denote by$v\inE^{0}$. For $\eta\in C_{d}(E^{1})$ andandfor any$f\in C_{0}(E^{0})$ , we definebyand$d=\mathrm{i}\mathrm{d}$$E^{1}$$C_{d}(E^{1})$$\langle\xi|\xi\rangle(v)=$$\langle\xi|\xi\rangle\in C_{0}(E^{0})$$\sum_{e\in d^{-1}(v)}|\xi(e)|^{2}<\infty$$\xi f\in C_{d}(E^{1})$$\xi,$$\langle\xi|\eta\rangle\in C_{0}(E^{0})$for e$(\xi f)(e)=\xi(e)f(d(e))$$\in E^{1}$,

for $v\in E^{0}$ .

$\langle \xi|\eta\rangle(v)=\sum_{e\in d^{-1}(v)}\overline{\xi(e)}\eta(e)$-module([Ka4, Proposition
is a(right) HilbertWith these operations,by $(\pi_{f}(f)\xi)(e)=f(r(e))\xi(e)$ foronof
1.10]). We define aleft action$e\in E^{1},$$f\in
C_{0}(E^{0})$.
. Thus we get aHilbert-bimodule
and$C_{0}(E^{0})$$C_{d}(E^{1})$$C_{0}(E^{0})$$\pi_{f}$$C_{d}(E^{1})$$C_{0}(E^{0})$$\xi\in C_{d}(E^{1})$$C_{d}(E^{1})$Definition 1.3 Let $E=(E^{0}, E^{1}, d, r)$ be acontinuous graph. AToeplitz $E$ -pair on aC’-homomorphism: $C_{0}(E^{0})arrow A$ isalgebra is apair of maps $T=(T^{0}, T^{1})$ where$C_{d}(E^{1})arrowA$
is alinear map satisfying that:
and$T^{0}$$A$$\mathrm{a}*$$T^{1}$

for(ii) $T^{0}(f)T^{1}(\xi)=T^{1}(\pi_{f}(f)\xi)$ for f(i)$T^{1}(\xi)’ T^{1}(\eta)=T^{0}(\langle\xi|\eta\rangle)$$\xi,$,
and$\eta\in C_{d}(E^{1})$$\in C_{0}(E^{0})$$\xi\in C_{d}(E^{1})$.

For $f\in C_{0}(E^{0})$ and $\xi\in C_{d}(E^{1})$ , the equation $T^{1}(\xi)T^{0}(f)=T^{1}(\xi f)$ holds automatically from the condition (i). For aToeplitz $E$-pair $T=(T^{0},T^{1})$ , we write $C’(T)$ forand . We can define
denoting the C’-algebra generated by the images of the mapsbyfor:-homomorphism$(;\in C_{d}(E^{1})$ .
for
by
is
definedwhere$T^{1}$$T^{0}$$\mathcal{K}(C_{d}(E^{1}))arrow C^{*}(T)$$\Phi^{1}$$\mathrm{a}*$$\xi,\eta\in C_{d}(E^{1})$$\Phi^{1}(\theta_{\xi,\eta})=T^{1}(\xi)T^{1}(\eta)’$$\theta_{\xi,\eta}(\zeta)=\xi\langle\eta|\zeta\rangle$$\theta_{\xi,\eta}\in \mathcal{K}(C_{d}(E^{1}))$

Definition 1.4 Let $E=(E^{0}, E^{1}, d, r)$ be acontinuous graph. We define three open
,
by
of
and
subsets$E_{\mathrm{s}\mathrm{c}\mathrm{e}}^{0},$$\mathrm{f}E_{\mathrm{i}\mathrm{n}}^{0}=$

and
and

{v

$\mathrm{f}E_{\mathrm{i}\mathrm{n}}^{0}$$E_{\mathrm{r}\mathrm{g}}^{0}$$E^{0}$$E_{\mathrm{s}\mathrm{c}\mathrm{e}}^{0}=E^{0}\backslash \overline{r(E^{1})}$

there exists aneighborhood V of v such that $r^{-1}(V)\subset E^{1}$ is compact},$\in E^{0}|$$E_{\mathrm{r}\mathrm{g}}^{0}=\mathrm{f}E_{\mathrm{i}\mathrm{n}}^{0}\backslash \overline{E_{\mathrm{s}\mathrm{c}\mathrm{e}}^{0}}$$E_{\mathrm{s}\mathrm{g}}^{0}=E^{0}\backslash E_{\mathrm{r}\mathrm{g}}^{0}$.

. We define two closed subsets$E_{\inf}^{0}$and$E_{\mathrm{s}\mathrm{g}}^{0}$of$E^{0}$

by$E_{\inf}^{0}=E^{0}\backslash \mathrm{f}E_{\mathrm{i}\mathrm{n}}^{0}$is adiscrete graph,is the set ofis called asource. When
Avertex inis the set of vertices which receive
vertices which receive finitely many edges, while
is said to be regular, and avertex inis said
infinitely many edges. Avertex in. We have thatandto be singular. Clearly we have that
Proposition
1.24]).
([Ka4,
Hence theand.
is an injection into
restriction of on$E$$E_{\mathrm{s}\mathrm{c}\mathrm{e}}^{0}$$\mathrm{f}E_{\mathrm{i}\mathrm{n}}^{0}$$E_{\inf}^{0}$$E_{8}^{0}\mathrm{g}$$E_{\mathrm{r}\mathrm{g}}^{0}$$E_{\mathrm{s}\mathrm{c}\mathrm{e}}^{0}\subset \mathrm{f}E_{\mathrm{i}\mathrm{n}}^{0}$$\mathrm{k}\mathrm{e}\mathrm{r}\pi_{f}=C_{0}(E_{\mathrm{s}\mathrm{c}\mathrm{e}}^{0})$$\pi_{f}$$E_{\mathrm{s}\mathrm{g}}^{0}=\overline{E_{\mathrm{s}\mathrm{c}\mathrm{e}}^{0}}\cup E_{\inf}^{0}$$\pi_{r}^{-1}(\mathcal{K}(C_{d}(E^{1})))=C_{0}(\mathrm{f}E_{\mathrm{i}\mathrm{n}}^{0})$$C_{0}(E_{\mathrm{r}\mathrm{g}}^{0})$$\mathcal{K}(C_{d}(E^{1}))$

Definition 1.5 Let $E=(E^{0}, E^{1}, d, r)$ be acontinuous graph. AToeplitz $E$-pair $T=$.
is called aCuntz-Krieger $E$ -pair if $T^{0}(f)=\Phi^{1}(\pi_{f}(f))$ for any
the universal C’-algebra generated by aCuntz-Krieger E-pairby
We denote.
$(T^{0}, T^{1})$$f\in C_{0}(E_{\mathrm{r}\mathrm{g}}^{0})$$\mathcal{O}(E)$75
is isomorphic to the graph algebra of the opposite
When $E$ is adiscrete graph,graph of $E$ . When acontinuous graph $E$ is defined by ahomeomorphism on alocallyZ. We
compact space $X,$
is isomorphic to the homeomorphism -algebra
have that is injective ([Ka4, Proposition 3.7]). Let be the group of complex numbers, there exists an action :with $|z|=1$ . By the universality of$\beta_{z}(t^{0}(f))=t^{0}(f)$.
for $f\in C_{0}(E^{0}),$ $\xi\in C_{d}(E^{1})$ andanddefined byinjectivity
of
says
that
the
theoremgauge
next
The action is called theaction. The
together with the existence of agauge action implies the universality of .
$\mathcal{O}(E)$$\sigma$$C^{*}$$\mathcal{O}(E)$$t^{0}$$C_{0}(X)\mathrm{x}_{\sigma}$$\mathrm{T}$$z\in \mathbb{C}$$\mathrm{T}\cap \mathcal{O}(E)$$\beta$$\mathcal{O}(E)$$z\in \mathrm{T}$$\beta_{z}(t^{1}(\xi))=zt^{1}(\xi)$$T^{0}$$\beta$$T$

Theorem 1.6 ([Ka4, Theorem 4.5]) For a continuous graph $E$ and a Cuntz-Krieger$E$ -pairis, the natural surjection
is an isomorphism if and only if$\beta_{z}’(T^{0}(f))=T^{0}(f)$$C^{*}(T)$andinjective and there exists an automorphism
such thatof$\beta_{z}’(T^{1}(\xi))=zT^{1}(\xi)$
every
.
for
$T$

$T^{0}$$\mathcal{O}(E)arrow C^{*}(T)$$\beta_{z}’$$z\in \mathrm{T}$2Invariant subsets of continuous graphs
We review definitions and results in [Ka5]. Let E $=(E^{0}, E^{1},$d, r) be acontinuous graph.

ofis said to be positively invaria. if $d(e)\in X^{0}$ implies$r(e)\in X^{0}$ for each, there exists, and to be negatively invariant if for$e\in E^{1}$ifis$r(e)=v$
invariantis said to beof. Asubset
withand ci(e)Definition 2.1 Asubset$X^{0}$$nt$$E^{0}$$e\in E^{1}$$v\in X^{0}\cap E_{\mathrm{r}\mathrm{g}}^{0}$$\in X^{0}$$X^{0}$$X^{0}$$E^{0}$

both positively and negatively invariant.

These terminologies coincides with the ordinal ones when continuous graphs are arising
is positively invariant if andfrom dynamical systems. When $E$ is adiscrete graph,only if its complement is hereditary, andis negatively invariant if and only if itsof ,complement is saturated (cf. [BHRS]). For aclosed positively invariant subset
we set $X^{1}=d^{-1}(X^{0})$ . Then $X=(X^{0}, X^{1}, d, r)$ is acontinuous graph. Aclosed positively.
invariant set
is invariant if and only if$X^{0}$$X^{0}$$X^{0}$$X^{0}$$X_{\mathrm{s}\mathrm{g}}^{0}\subset E_{8}^{0}\mathrm{g}\cap X^{0}$

Definition 2.2 Apair $\rho=(X^{0}, Z)$ of closed subsets ofconditions is called an admissible pair;
(i)

(ii)

$X^{0}$$E^{0}$$E^{0}$

satisfying the following two

is invariant,.

$X_{\mathrm{s}\mathrm{g}}^{0}\subset Z\subset E_{\mathrm{s}\mathrm{g}}^{0}\cap X^{0}$

Definition 2.3 For an admissible pair $\rho=(X^{0}, Z)$ , we define acontinuous graph, and define
as follows. Set$(E_{\rho}^{0}, E_{\rho}^{1}, d_{\rho}, r_{\rho})$$\mathrm{Y}_{\rho}=X_{\mathrm{r}\mathrm{g}}^{0}\cap Z,$$E_{\rho}^{0}=X^{0}\mathrm{I}\mathrm{I}_{\partial \mathrm{Y}_{\rho}}\overline{\mathrm{Y}_{\rho}}$

The domain map
range map :$r_{\rho}$$d_{\rho}$:$\partial \mathrm{Y}_{\rho}=\overline{\mathrm{Y}_{\rho}}\backslash \mathrm{Y}_{\rho}$$E_{\rho}^{1}=X^{1}\mathrm{u}_{d^{-1}(\partial \mathrm{Y}_{\rho})}d^{-1}(\overline{\mathrm{Y}_{\beta}})$is defined from $d:X^{1}arrow X^{0}$ andis defined from $r:X^{1}arrow X^{0}$ and$E_{\rho}^{1}arrow E_{\rho}^{0}$$E_{\rho}^{1}arrow E_{\rho}^{0}$,$E_{\rho}=$.

$d:d^{-1}(\overline{\mathrm{Y}_{\rho}})arrow\overline{\mathrm{Y}_{\rho}}$$r:d^{-1}(\overline{\mathrm{Y}_{\rho}})arrow X^{0}$

.

. The

76
Note that for an admissible pair $\rho=(X^{0},$Z) with Z-homomorphism
and:a -subalgebra$=X_{\mathrm{r}\mathrm{g}}^{0}$$C^{*}$$\mathcal{F}^{1}\subset \mathcal{O}(E)$$\pi_{0}^{1}$$\mathrm{a}*$, we have $E_{\rho}=X$ . Defineby$\mathcal{F}^{1}arrow C_{0}(E_{\mathrm{s}\mathrm{g}}^{0})$$\mathcal{F}^{1}=\{t^{0}(f)+\varphi^{1}(x)|f\in C_{0}(E^{0}),x\in \mathcal{K}(C_{d}(E^{1}))\}$

andof$\pi_{0}^{1}(t^{0}(f)+\varphi^{1}(x))=f|_{E_{*\mathrm{g}}^{0}}$$Z_{I}$$E^{0}$. For an ideal I of$\mathcal{O}(E)$,, we define closed subsets$X_{I}^{0}$andby$X_{I}^{0}=$$Z_{I}=${ v $\in E^{0}|f(v)=0$ for all f
for all f{v$\in E_{8}^{0}\mathrm{g}|f(v)=0$

Proposition 2.4 For an ideal I of$\mathcal{O}(E)$$\in C_{0}(E^{0})$with $t^{0}(f)\in I$ },$\in\pi_{0}^{1}(I\cap \mathcal{F}^{1})$, the pair}.

$\rho_{I}=(X_{I}^{0}, Z_{I})$is an admissible pair.

By using Theorem 1.6, we can show the following.
Proposition 2.5 For a gauge-invariant ideal I ofphism.

$\mathcal{O}(E)$, there exists a natural isomor-$\mathcal{O}(E)/I\cong \mathcal{O}(E_{\beta I})$

From this proposition and some computation, we get the next theorem.
gives us an inclusion reversing one-tO-One corresponTheorem 2.6 The map I
gauge-invariant
ideals and the set of all admissible pairs.
dence betrneen the set of all$1arrow\rho_{I}$This theorem is acontinuous counterpart of [BHRS, Theorem 3.6]. It is known thatgauge-invariant ideals of ahomeomorphism C’-algebra correspond bijectively to closed
invariant subsets [T2, Theorem 2]. The next proposition is ageneralization of this fact.

, the map
Proposition 2.7 When a continuous graph $E$ satisfies thatgives an inclusion reversing one-tO-One correspondence betrneen the set of all gaugeinvariant ideals and the set of closed invariant sets.
$I\vdasharrow$$E_{\mathrm{r}\mathrm{g}}^{0}=E^{0}$$X_{I}^{0}$, we have. Hence admissible
pairs correspond bijectively to closed invariant subsets. Now the assertion follows from
Theorem 2.6.
1

Proof. For aclosed invariant set$X^{0}$$X_{\mathrm{s}\mathrm{g}}^{0}=E_{8}^{0}\mathrm{g}\cap X^{0}=\emptyset$

3Free and topologically free continuous graphs
For $n=2,3,$$\ldots$, we define aspace$E^{n}$of paths with length

$n$

by

$E^{n}=\{(e_{n}, \ldots, e_{2}, e_{1})\in E^{1}\cross\cdots \mathrm{x}E^{1}\cross E^{1}|d(e_{k+1})=r(e_{k})(1\leq k\leq n-1)\}$.

We define domain and range maps $r:E^{n}arrow E^{0}$ by $d(e)=d(e_{1})$ and $r(e)=r(e_{n})$ for$e=(e_{n}, \ldots, e_{1})\in E^{n}$ . Apath $e=(e_{n}, \ldots, e_{1})\in E^{n}(n\geq 1)$ is called aloop if $r(e)=d(e)$ ,and the vertex $r(e)=d(e)$ is called the base point of the loop . Aloop $e=(e_{n}, \ldots, e_{1})$is said to be without entrances if $r^{-1}(r(e_{k}))=\{e_{k}\}$ for $k=1,$
.
$d,$

$e$

$\ldots,$$n$

77
Definition 3.1 Acontinuous graph $E$ is said to be topologically free if the set of basepoints of loops without entrances has an empty interior.

This generalizes topological freeness of ordinary dynamical systems and Conditionof graph algebras (see, for example, [T1] and [KPR]).

$\mathrm{L}$Theorem 3.2 ([Ka4, Theorem 5.12]) If a continuous graph $E=(E^{0}, E^{1}, d, r)$ is topOis an isomorphism for all Cuntzlogically free, then the natural surjection
is injective.
Krieger $E$ -pair $T=(T^{0}, T^{1})$ such that$\mathcal{O}(E)arrow C^{*}(T)$$T^{0}$

By the above theorem, we have the following (cf. Proposition 2.5).

Proposition 3.3 ([Ka5]) Let I be an ideal of
logically free, then I is gauge-invariant.

We define apositive orbit space$\mathrm{O}\mathrm{r}\mathrm{b}^{+}(v)=\{v\}\cup$$\mathcal{O}(E)$$\mathrm{O}\mathrm{r}\mathrm{b}^{+}(v)\subset E^{0}$. If a continuous graph$E_{\beta I}$

is topO-of $v\in E^{0}$ by{ $r(e)\in E^{0}|e\in E^{n}$ with $d(e)=v(n\geq 1)$ }.

is positively invariant if and only ifof
It is easy to see that asubset$v\in E^{0}$ ,$v\in X^{0}$$L(v)\subset
E^{0}$ by
we define. For
for all$X^{0}$$E^{0}$$L(v)=\{v’\in \mathrm{O}\mathrm{r}\mathrm{b}^{+}(v)|v\in \mathrm{O}\mathrm{r}\mathrm{b}^{+}(v’)\}$

Definition 3.4 For apositive integer , we denote by
the following three conditions;$n$$\mathrm{O}\mathrm{r}\mathrm{b}^{+}(v)\subset X^{0}$.

$\mathrm{P}\mathrm{e}\mathrm{r}_{n}(E)$the set of vertices$v_{1}$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\Phi \mathrm{i}\mathrm{n}\mathrm{g}$(i) $L(v_{1})$ is afinite set$\{v_{1}, v_{2}, \ldots, v_{n}\}$,(ii) $\{e\in E^{1}|d(e), r(e)\in L(v_{1})\}=\{e_{1}, e_{2}, \ldots, e_{n}\}$ with $d(e:)=v$:and $r(e:)=v:+1$ for$i=1,2,$where $v_{n+1}=v_{1}$ ,$\ldots,$(iii)$v_{1}$$n$is isolated in

We set Per(E)$\mathrm{O}\mathrm{r}\mathrm{b}^{+}(v_{1})$$= \bigcup_{n=1}^{\infty}\mathrm{P}\mathrm{e}\mathrm{r}_{n}(E)$.
and Aper(E)$=E^{0}\backslash \mathrm{P}\mathrm{e}\mathrm{r}(E)$.

An element in Per(E) is called aperiodic point while an element in Aper(E) is called

an aperiodic point.

Definition 3.5 Acontinuous graph $E$ is said to be free if Aper(E) $=E^{0}$ .
This is ageneralization of freeness of ordinary dynamical systems and Condition K ofgraph algebras (see, for example, [KPRR]).

Proposition 3.6 ([Ka5]) A continuous graph $E$ is free if and only ifffee for every admissible pair .

$E_{\rho}$

is topologically$\rho$

In particular, free continuous graphs are topologically free.
Theorem 3.7 ([Ka5]) If a continuous graph $E$ is free, then every ideal is gauge-invariant.
Hence the set of all ideals corresponds bijectively to the set of all admissible pairs by themap.
$I\vdash\neq\rho_{I}$Proof. Clear from Proposition 3.6, Proposition 3.3 and Theorem 2.6.

1

78
Crossed products of Cuntz algebras

4

For $n=2,3,$
isometries$\infty$$\ldots,$$S_{1},$ $S_{2},$, the Cuntz algebrais the universal C’-algebra generated by(we also use this notation for $n=\infty$ ), satisfying$\mathcal{O}_{n}$$n$$S_{n}$$\ldots,$$\sum_{\dot{l}=1}^{n}S_{\dot{l}}S_{\dot{l}}^{*}=1$(for any i,j with i$S_{\dot{l}}^{*}S_{j}=0$$\neq j$)if n$<\infty$,

if n$=\infty$.

whose dual group is denoted by . We
We fix alocally compact abelian group
multiplicative operations of abelian groups except for T. The pairing ofalways$t\in G$ and.
is denoted by$G$$\Gamma$$\mathrm{u}\mathrm{s}\mathrm{e}+\mathrm{f}\mathrm{o}\mathrm{r}$$\gamma\in\Gamma$$\langle t|\gamma\rangle\in \mathrm{T}$Definition 4.1 Letby$\omega=(\omega_{1},\omega_{2}, \ldots, \omega_{n})\in\Gamma^{n}$be given. We define the action$\alpha^{\omega}$: $G\cap$$\mathcal{O}_{n}$$(i=1,2, \ldots,n, t\in G)$$\alpha_{t}^{\iota v}(S_{\dot{l}})=\langle t|\omega:\rangle S_{\dot{l}}$.

,by the action
We recall some elementary facts on the crossed productproduct,
has aC’-subalgebra
which was stated in [Kal]. The crossed
via the Fourier transform. We denote by the isomorphism
which is isomorphic to$\alpha^{\omega}$$\mathcal{O}_{n}\mathrm{x}_{\alpha^{\omega}}G$$\mathcal{O}_{n^{\aleph}\alpha^{\omega}}G$$\mathbb{C}1\aleph_{\alpha^{\omega}}G$$T^{0}$$C_{0}(\Gamma)$$T^{0}$:$C_{0}(\Gamma)arrow \mathbb{C}1\nu_{\alpha^{\omega}}G\subset \mathcal{O}_{n}\mathrm{x}_{\alpha^{\omega}}G$.

is naturally embedded into the multiplier algebra
The Cuntz algebra
is generated as aC’-algebra by
G. The crossed product$\mathcal{O}_{n}$of$\mathcal{O}_{n}\aleph_{\alpha^{\omega}}G$$\mathcal{O}_{n}\cross_{\alpha^{\omega}}$$\{S_{}T^{0}(f)|i\in\{1, \ldots, n\}, f\in C_{0}(\Gamma)\}$, we define a(reverse) shift automorphism. Then we havefor allthe gauge action of , we can define an action 73:
gauge action. We have $\beta_{z}(T^{0}(f))=T^{0}(f)$ and$i\in\{1, \ldots,n\}$ , and.
For$M(\mathcal{O}_{n}\mathrm{n}_{\alpha^{\omega}}G)$$\gamma_{0}\in\Gamma$$T^{0}(f)S_{\dot{l}}=S_{1}.T^{0}(\sigma_{\omega_{}}f)$$f(\gamma+\gamma_{0})$$\mathcal{O}_{n}$$\sigma_{\gamma 0}$:.

$C_{0}(\Gamma)arrow C_{0}(\Gamma)$$f\in C_{0}(\Gamma)$by$(\sigma_{\gamma 0}f)(\gamma)=$and $i\in\{1, \ldots, n\}$ . From
which is also called a,for$\mathrm{T}\cap \mathcal{O}_{n}\mathrm{x}_{\alpha^{\omega}}G$$\beta_{z}(S_{\dot{l}}T^{0}(f))=zS_{\dot{l}}T^{0}(f)$$f\in C_{0}(\Gamma)$$z\in \mathrm{T}$Definition 4.2 Let$E_{\omega}=(E_{\omega}^{0}, E_{\omega}^{1}, d_{\omega},r_{\omega})$$i=1,2,$map$r_{\omega}$$\ldots,$

:

Each
and

$n$

as follows. We set. The map

$E_{\omega}^{1}arrow E_{\omega}^{0}$$v\in E_{(d}^{0}$

be given. We define acontinuous graph
and
where
for
is defined by identity maps on each , and thefor $i=1,2,$.

$\omega=(\omega_{1},\omega_{2}, \ldots,\omega_{n})\in\Gamma^{n}$:is defined by$d_{\omega}$$E_{\omega}^{1}arrow E_{\omega}^{0}$$E_{\omega}^{0}=\Gamma$$E_{\iota v}^{1}= \prod_{\dot{l}=1}^{n}\mathrm{r}_{:}$$r_{\omega}|_{\Gamma:}(\gamma)=\gamma+\omega_{i}$$\ldots,$$n$receives and emits -edges. It is easy to see that
if $n=\infty$ . Since
is defined by identity maps, we have$n$$E_{\omega}^{0}=(E_{\omega}^{0})_{\inf}$$E_{\omega}^{0}=(E_{\omega}^{0})_{\mathrm{r}\mathrm{g}}$has natural Hilbert
satisfies$C_{0}(\mathrm{r}_{:})=C_{0}(\Gamma)$$C_{0}(\Gamma)arrow \mathcal{L}(C_{d_{\omega}}(E_{\omega}))$if $n<\infty$ ,$d_{\mathrm{t}\theta}$$C_{d_{\omega}}(E_{\omega})=\oplus^{n}C_{0}(\mathrm{r}_{:})\dot{l}=1$

where$\mathrm{r}_{:}=\Gamma$$\Gamma_{i}$$C_{0}(\Gamma)$’-module structure. The left action$\pi_{r_{\omega}}(f)(\xi_{1}, \xi_{2}, \ldots, \xi_{n})=(\sigma_{\omega_{1}}(f)\xi_{1}, \sigma_{\omega_{2}}(f)\xi_{2},$$\ldots,$$\sigma_{\omega_{n}}(f)\xi_{n})\in\oplus^{n}C_{0}(\mathrm{r}_{:})\dot{l}=1$’$\pi_{t_{\omega}}$

:

79
for f
and
:
We have a -homomorphism
by$\in C_{0}(\Gamma)$$(\xi_{1}, \xi_{2},$\ldots.

$\xi_{n})\in\oplus_{i=1}^{n}C_{0}(\Gamma_{i})$:$T^{0}$$*$$C_{0}(\Gamma)arrow \mathcal{O}_{n}\mathrm{x}_{\alpha^{\omega}}G$. We define alinear map$T^{1}$:$\oplus_{i=1}^{n}C_{0}(\Gamma_{i})arrow \mathcal{O}_{n}\mathrm{x}_{\alpha^{\omega}}G$$T^{1}(\xi_{1}, \xi_{2},$for$(\xi_{1},\xi_{2},$\ldots ,\ldots ,$\xi_{n})\in\oplus_{\dot{l}=1}^{n}C_{0}(\mathrm{r}_{:})$$\xi_{n})=\sum_{i=1}^{n}S_{i}T^{0}(\xi_{i})\in \mathcal{O}_{n}\mathrm{x}_{\alpha^{\omega}}G$.

Proposition 4.3 The pair $T=(T^{0}, T^{1})$ is a Cuntz-Krieger
isomorphism.

-pair because, we see that

$E_{\omega}$$f\in C_{0}(\Gamma)$-pair, and this induces an$\mathcal{O}(E_{\omega})\cong \mathcal{O}_{n\alpha^{\omega}}\aleph G$

Proof. It is not difficult to see that

Krieger$E_{\omega}$$T$

is aToeplitz -pair. When $n=\infty,$. When $n<\infty$ , we have
$E_{\omega}$$C_{0}((E_{\omega}^{0})_{\mathrm{r}\mathrm{g}})=0$$T$

is aCuntz. For$C_{0}((E_{\omega}^{0})_{\mathrm{r}\mathrm{g}})=C_{0}(\Gamma)$,$\pi_{\mathrm{r}_{\omega}}(f)=\sum_{\dot{l}=1}^{n}\theta_{\xi\eta:}:$where$\xi_{\dot{l}},$$\eta_{\dot{*}}\in C_{0}(\mathrm{r}_{:})$

satisfies that$\xi_{1}.\overline{\eta_{\dot{*}}}=\sigma_{\omega}.\cdot(f)$for $i=1,2,$$\ldots,$$n$. We have.

$\Phi^{1}(\pi_{\gamma_{\omega}}(f))=\sum_{i=1}^{n}T^{1}(\xi_{\dot{l}})T^{1}(\eta_{\dot{l}})^{*}=\sum_{\dot{\iota}=1}^{n}S_{i}T^{0}(\xi_{\dot{l}})T^{0}(\eta_{i})^{*}S_{1}^{*}$.

$=. \sum_{1=1}^{n}S_{\dot{l}}T^{0}(\sigma_{\omega}.\cdot(f))S_{1}^{*}$

Hence

on

$T$

$=. \sum_{1=1}^{n}T^{0}(f)S_{\dot{l}}S_{1}.’=T^{0}(f)$.

is aCuntz-Krieger -pair. By definition,is injective, and the gauge action
satisfies the condition of Theorem 1.6. Hence the natural surjection
is an isomorphism.
1$T^{0}$$E_{\omega}$$\mathcal{O}_{n}\mathrm{n}_{\alpha^{\omega}}G$$\mathcal{O}_{n}\mathrm{x}_{\alpha^{\omega}}G$$\mathcal{O}(E_{\omega})arrow$

5Ideal structures of(n $<\infty)$$\mathcal{O}_{n}\mathrm{x}_{\alpha^{\omega}}G$

In this section, we discuss the ideal structure ofin the case that $n<\infty$ . Let
be an integer grater than 1, and take
. In [Kal], we introduced the following
notion.
$\mathcal{O}_{n}n_{\alpha^{\omega}}G$$n$$\omega\in\Gamma^{n}$Definition 5.1 ([Kal, Definition 3.2]) Asubset
aclosed set satisfying the following two conditions:(i) For any$\gamma\in X^{0}$(ii) For any$\gamma\in X^{0}$$X^{0}$

of$\Gamma$

is called -invariant if{1,2,n}, we have
, there exists i {1,2,n} such thatand any i$\in$$\gamma+\omega:\in X^{0}$\ldots ,$\in$$\omega$\ldots ,$X^{0}$.
, and the-invariant set$X^{0}\subset\Gamma=E^{0}$$X^{0}$

is.

$\gamma^{-\omega}:\in X^{0}$

The condition (i) above corresponds to positive invariance ofcondition (ii) corresponds to negative invariance of. Hence$X^{0}$

is an$\omega$

80
is aclosed invariant set of the continuous graphif and only ifby, we define$X^{0}$$E_{\omega}$. For an ideal I of

$X_{I}^{0}\subset\Gamma$$\mathcal{O}_{n}\mathrm{x}_{\alpha^{\omega}}G$$X_{I}^{0}=${is an -invariant subset ofThenof main results in [Kal].
$X_{I}^{0}$for all$\gamma\in\Gamma|f(\gamma)=0$$\Gamma$$\omega$$f\in C_{0}(\Gamma)$with $T^{0}(f)\in I$ }.

([Kal, Proposition 3.3]). The following is the one

Theorem 5.2 ([Kal, Theorem 3.14]) The correspondence
reversing bijection between the set of gauge-invariant ideals of-invariant subsets of .

$I\vdash*X_{I}^{0}$gives an inclusion
and the set of$\mathcal{O}_{n}\mathrm{x}_{\alpha^{\omega}}G$$\Gamma$$\omega$1

Proof. This follows from Theorem 2.6 and Proposition 2.7.

Definition 5.3 ([Kal, Definition 4.2]) An -invariant subset $X$ of is said to besuch that there is only one element $i_{0}\in\{1,2, \ldots, n\}$ with
bad if there exists, and this element satisfies that $m\omega:_{0}=0$ for some positive integer $m$ . An
-invariant subset $X$ of is said to be good if $X$ is not bad.
$\Gamma$$\omega$$\gamma_{0}\in X$$i_{0}$$\gamma_{0}-\omega_{\dot{l}_{0}}\in X$$\Gamma$$\omega$Lemma 5.4 An ci-invariant subset$(X^{0}, X^{1}, d, r)$ is topologicallyfree.

$X^{0}$

is good if and only if the continuous graph $X=$

that there
is bad, then there exists-invariant subset$i_{0}\in\{1,2,\ldots,n\}$for some positive
andwithis only one element. The set is an open subset of and it is not
integer $m$ . Let. All
is abase point of aloop
empty because

Proof. If an$X^{0}$$\omega$$\gamma_{0}\in X^{0}\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\Phi \mathrm{i}\mathrm{n}\mathrm{g}$$\gamma_{0-\omega}:_{0}\in X^{0}$$m\omega_{\dot{l}_{0}}=0$$X^{0}$$V$$V=X^{0} \backslash \bigcup_{\dot{l}\neq\dot{l}0}X^{0}+\omega:$$\gamma\in V$$\gamma_{0}\in V$.
$\gamma_{arrow^{\gamma+\omega_{0}}arrow\cdots\cdotsarrow^{\gamma+m\omega_{0}=\gamma}}..\cdot..\cdot$

which has no entrances in the continuous graph $X$ . Hence the continuous graph $X$ is
not topologically free. Conversely if the continuous graph $X$ is not topologically free,then abase point 7of aloop without entrances satisfies that there is only one element$i_{0}\in\{1,2, \ldots, n\}$ with, and for some positive integer $m$ we have $m\omega:_{0}=0$ .
is bad.
Hence$\gamma_{0-\omega}:_{0}\in X^{0}$$X^{0}$$\mathrm{I}$

Proposition 5.5 ([Kal, Theorem 4.5]) Let I be an ideal of
good. Then I is gauge-invariant.

$\mathcal{O}_{n}\aleph_{\alpha^{\omega}}G$

such that$X_{I}^{0}$

is

1

Proof. Combine Proposition 3.3 and Lemma 5.4.

is said to satisfy Condition 5.1 if for each $i\in\{1,2, \ldots, n\}$ , one of
An element
the following two conditions is satisfied ([Kal]):.
(i) For any positive integer
is in the closed semigroup generated by(ii) There exists $j\neq i$ such$\omega\in\Gamma^{n}$$k,$$k\omega:\neq 0$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}-\omega_{j}$$\mathrm{a}\mathrm{n}\mathrm{d}-\omega:$$\omega_{1},$$\ldots,\omega_{n}$.

It is not difficult to see that Condition 5.1 is exactly same as the condition that ais free. Hence from Theorem 3.7, we get the following.
continuous graph
Proposition 5.6 ([Kal, Theorem 5.2]) When satisfies Condition 5.1, all ideals aregauge-invariant and there is one-tO-One correspondence between the set of ideals ofand the set of -invariant subsets of .
$E_{\omega}$$\omega$$a$$\mathcal{O}_{n}\mathrm{x}_{\alpha^{\omega}}G$$\Gamma$$\omega$

81
6Ideal structures of$\mathcal{O}_{\infty}\mathrm{n}_{\alpha^{\alpha)}}G$. The argument
In [Ka3], we discussed, among others, the ideal structure of$n<\infty$need to change
wedone in [Kal]. However
there was analogous to the case that”some details, for example, the definition of -invariant sets. Takeand fix it.
is
of is called -invariant if
Definition 6.1 ([Ka3, Definition 3.3]) Asubset
for any positive integer .
aclosed set with$\mathcal{O}_{\infty}\mathrm{x}_{\alpha^{\omega}}G$$\omega=(\omega_{1},\omega_{2}, \ldots)\in \mathrm{I}$$\omega$$X^{0}$$X^{0}+\omega:\subset X^{0}$$\Gamma$$X^{0}$$\omega$$i$

An -invariant set is same as aclosed positively invariant set in the continuous graphare invariant because. However, note that every positively invariant subsets of. Hence we see that -invariant sets are same as closed invariant sets. For anby, we define aclosed set-invariant set$\omega$$E_{\omega}$$E_{\omega}$$(E_{\omega}^{0})_{\mathrm{r}\mathrm{g}}=\emptyset$$\omega$$\omega$$X^{0}$$H_{X^{0}}$$H_{X^{0}}=X^{0} \backslash \bigcup_{1}(X^{0}+\omega:=\infty)\cup n=1|.=n\cap\cup(X^{0}+\omega_{1})\subset X^{0}\infty\infty\cdot$.

of subsets ofDefinition 6.2 ([Ka3, Definition 3.4]) Apairsatispingis aclosed set-invariant if
is an -invariant set, and
It is not difficult to see that$\overline{X}=(X^{0}, X^{\infty})$$\omega$$X^{0}$$X_{\epsilon \mathrm{c}\mathrm{e}}^{0}=X^{0}\backslash \cup(X^{0}+\omega:\infty)$,

is called$H_{X^{0}}\subset X^{\infty}\subset X^{0}$$X^{\infty}$$\omega$$\Gamma$$X_{\inf}^{0}=\cap\cup(X^{0}+\omega:\infty\infty)$.

,$n=1:=n$$:=1$. From this fact, we see that the definition of u-invariant
and
, we
and
pairs is same as the one of admissible pairs. For an ideal I ofof bydefine the closed subset$H_{X^{0}}=\overline{X_{\mathrm{s}\mathrm{c}\mathrm{e}}^{0}}\cup X_{\inf}^{0}=X_{\mathrm{s}\mathrm{g}}^{0}$$\mathcal{O}_{\infty}\nu_{\alpha^{\omega}}G$$X_{I}^{n}$$X_{I}^{n}=${$n\in \mathrm{N}$$\Gamma$$\gamma\in\Gamma|f(\gamma)=0$

where $P_{0}=1$ andas in Section 2. Set
Proposition 3.5]). We can see thatwith $P_{n}T^{0}(f)\in I$ },is same. Clearly, the definition ofis -invariant([Ka3,The pair. Hence Theorem 2.6 gives the following.

for all$f\in C_{0}(\Gamma)$$X_{I}^{0}\subset\Gamma$$P_{n}=1- \sum_{1=1}^{n}.S_{1}.S_{}’\in \mathcal{O}_{\infty}$$X_{I}^{\infty}= \bigcap_{n=0}^{\infty}X_{I}^{n}$.

$\overline{X}_{I}=(X_{I}^{0},X_{I}^{\infty})$$\omega$$X_{I}^{\infty}=Z_{I}$gives a bijection
Theorem 6.3 ([Ka3, Theorem 3.16]) The correspondence I
and the set of -invariant pairs.
between the set of gauge-invariant ideals of”, one of theis said to satisfy Condition 5.1 if for each
An element
following two conditions is satisfied:.
(i) For any positive integer$i_{n_{h},k}(n_{k}\geq 1)$ with, there exist positive integers(ii) For $k=1,2,$.
and
Similarly as in the case of $n<\infty$ , we see that Condition 5.1 is exactly same as theis free. Hence from Theorem 3.7, we get thecondition that acontinuous graphfollowing.
Theorem 6.4 ([Ka3, Theorem 5.3]) Suppose that satisfies Condition 5.1. Then all
is gauge-invariant. Hence there exists one-tO-One correspondence
ideal ofand the set of -invariant pairs of subsets of .
betrneen the set of ideals of$\mathrm{I}arrow\tilde{X}_{I}$$\mathcal{O}_{\infty}\mathrm{n}_{\alpha^{\omega}}G$$\omega$$i\in \mathbb{Z}_{+}$$\omega\in \mathrm{I}$$k,$$k\omega:\neq 0$$i_{1,k},$$\ldots$$\ldots,$$i_{1fl}\neq i$$\lim_{karrow\infty}\sum_{j=1}^{n_{k}}\omega_{\dot{l}_{\mathrm{j}.k}}=0$$E_{\omega}$$\omega$$a$$\mathcal{O}_{\infty}\aleph_{\alpha^{\omega}}G$$\Gamma$$\mathcal{O}_{\infty}\mathrm{x}_{\alpha^{\omega}}G$$\omega$82
7Primitive ideal spaces
by using primitive ideal
In [Kal] and [Ka3], we studied the ideal structures ofspaces when does not satisfy Condition 5.1. These works can be considered as continuous
counterparts of [HS]. So far, the author has not succeeded in generalizing these results
to more general continuous graphs which are not free. Note that acontinuous graphdefined here is aspecial kind of continuous graph which satisfies that every vertices receiveand emit same number of edges in the same way.
$\mathcal{O}_{n}x_{\alpha^{\omega}}G$$\omega$$E_{\omega}$

References[BHRS] Bates, T.; Hong, J.; Raeburn, I.;
algebras of infinite graphs. Preprint.

, W. The ideal structure of the$\mathrm{S}\mathrm{z}\mathrm{y}\mathrm{m}\mathrm{a}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}$$C^{*}-$[FLR]

Fowler, N. J.; Laca, M.; Raeburn, I. The C’-algebras of infinite graphs. Proc.
Amer. Math. Soc. 128 (2000), no. 8, 2319-2327.

[HS]

Hong, J. H.; Szymanski, W. The primitive ideal space of the C’-algebras ofinfinite graphs. Preprint.

[Kal]

Katsura, T. The ideal structures of crossed products of Cuntz algebras by quasifree actions of abelian groups. To appear in Canad. J. Math.

[Ka2]

Katsura, T. $AF$-embeddability of crossed products of Cuntz algebras. To appear
in J. Funct. Anal.

[Ka3]

by quasi-free actions of
Katsura, T. On crossed products of the Cuntz algebraabelian groups. To appear in the proceedings of OAMP Conference (Constanta
2001).

[Ka4]

Katsura, T. A class of C’-algebras generalizing both graph algebras and homeOmorphism C’-algebras I, fundamental results. Preprint.

[Ka5]

Katsura, T. A class of C’-algebras generalizing both graph algebras and homeOmorphism C’-algebras III, ideal structures. Preprint.

[KPR]

Kumjian, A.; Pask, D.; Raeburn, I. Cuntz-Krieger algebras of directed graphs.
Pacific J. Math. 184 (1998), no. 1, 161-174.

$\mathcal{O}_{\infty}$[KPRR] Kumjian, A.;Pask, D.; Raeburn, I.; Renault, J. Graphs, groupoids, and CuntzKrieger algebras. J. Funct. Anal. 144 (1997), no. 2, 505-541.

[T1]

Tomiyama, J. The interplay betrneen topological dynamics and theory of C’algebras. Lecture Notes Series, 2. Seoul National University, Research Instituteof Mathematics, Global Analysis Research Center, Seoul, 1992.

[T2]

Tomiyama, J. Stmcture of ideals and isomorphisms of C’-crossed products bysingle homeomorphism. Tokyo J. Math. 23 (2000), no. 1, 1-13.

83
[T3]

Tomiyama, J. Hulls and kernels from topological dynamical systems and theirapplications to homeomorphism C’-algebras. Preprint.

[T4]

Tomiyama, J. On the projection theorem for homeomorphism C’-algebras.
Preprint.