SIMPLE CROSSED PRODUCTS OF $C^{*}$ -ALGEBRAS BY ...

YOKOHAMA MATHEMATICAL JOURNAL VOL. 28, 1980

SIMPLE CROSSED PRODUCTS OF -ALGEBRAS BY LOCALLY COMPACT ABELIAN GROUPS $C^{*}$

By

AKITAKA KISHIMOTO (Received Dec. 25, 1979)

ABSTRACT: We introduce a new invariant , a closed subsemigroup of the dual of , of a $c*$ -dynamical system where is a -algebra, and is a locally compact abelian group with an action on . We show that the crossed product $a\times.G$ is simple $\tilde{\Gamma}(\alpha)$

$G$

$(\mathfrak{a}, G, \alpha)$

$\alpha$

$\mathfrak{a}$

$C^{*}$

$G$

$\mathfrak{a}$

if and only if is -simple (i.e. does not have any non-trivial a-invariant ideals) and equals the dual of $G$ . We discuss some cases where coincides with the Connes spectrum . Finally we give examples of simple crossed products of Cuntz algebras by locally compact abelian groups. $\mathfrak{a}$

$\alpha$

$\tilde{\Gamma}(\alpha)$

$\mathfrak{a}$


$\Gamma(\alpha)$

\S 1. Introduction In a paper [7] by D. Olesen and G. K. Pedersen the Connes spectrum [6] of a -dynamical system , with a locally compact abelian group $G$ , plays an important role in characterizing primeness of the crossed product . We introduoe a new invariant , which is a closed subsemigroup of , and show that is relevant in characterizing simplicity of . After being introduced, our results and methods are quite similar to the prime case above-mentioned. By using a characterization of in terms of ideals of the crossed product and the dual action on it, we show that coincides with in some cases, in particular, when is discrete and a is -simple. Unfortunately seems to be hard to compute, at least, directly from its definition. Hence our examples of simple crossed products could be given independently of the above-mentioned general theory. We show that the crossed product of a Cuntz algebra $O_{n}[3]$ by a so-called quasi-free automorphism group [4] corresponding to a unitary representation , on the n-dimensional Hilbert space, of a locally compact abelian group , is simple if and only if the closed subsemigroup generated by Sp and $-p$ equals the dual of , for any $p\in Spu$ if $ n\neq e_{1}$

$e(\Omega_{1})$

$q$

$\Omega_{1}$

$[\pi(\mathfrak{a})e(\Omega_{1})\ovalbox{\tt\small REJECT}_{\pi}]=\ovalbox{\tt\small REJECT}_{\pi}$

sinoe the projection onto the left hand side is in

$\pi(\mathfrak{a})^{\prime}\cap u(G)^{\prime}$

.

Since

$e[\pi(\mathfrak{a})e(\Omega_{1})\ovalbox{\tt\small REJECT}_{\pi}]=[\pi(B)e(\Omega_{1})\ovalbox{\tt\small REJECT}_{\pi}]$

$[\pi(B^{\alpha}(\Omega))e(\Omega_{1})\ovalbox{\tt\small REJECT}_{\pi}]=(0)$

for any compact neighbourhood

$\Omega_{1}$

of , we can easily conclude that Sp ue$ $p+q$ . $q$

$q.e.d$

.

Remark 2.3. The intersection over the covariant representations can be restricted to the irreducible covariant representations $(\pi, u)$ , i.e. the ones with , in the above proposition. $\pi(a)^{\prime}\cap$

$u(G)^{\prime}=C\cdot 1$

, it follows easily that Remark 2.4. If is not simple (c.f. [9]). We may consider the invariant in von Neumann algebra case corresponding $F$ to , but it turns out to be the Connes spectrum. Hence $F$ can be considered as another version of the Connes spectrum (originally defined for von Neumann $ F(\alpha)\neq\Gamma$

algebras).

$\mathfrak{a}\times_{\alpha}G$

AKITAKA KISHIMOTO

72

\S 3. Simple crossed products -dynamical system with a locally compact abelian group Let be a $G$ . Let $K\equiv C(L^{2}(G))$ be the algebra of all compact operators on $L^{2}(G)$ and the regular representation of on $L^{2}(G)$ . Let and an action of $C^{*}$


$\lambda$

$G$

$\tilde{\alpha}=\alpha\otimes Ad\lambda$

$\tilde{\mathfrak{a}}=\mathfrak{a}\otimes K$

on .

$G$

$\tilde{\mathfrak{a}}$

Lemma 3.1.

$F(\tilde{\alpha})=\tilde{\Gamma}(\alpha)$

.

. Then there are a covariant representation $(\pi, u)$ Proof. Suppose and such that Sp $ue\ni O$ and Sp (the where is the identity of $ue\ni O$ assumption Sp is always achieved by multiplying by some character of ). We construct a covariant representation of by simply tensoring $K,$ , Ad ), e.g. the identity representation of ( $p\not\in F(\alpha)$

$B\in\ovalbox{\tt\small REJECT}^{\alpha}(\mathfrak{a})$

$ue\Rightarrow p$

$\overline{\pi(B})^{w}$

$e$

$G$

$u$

$(\tilde{\pi},\tilde{u})$

$G$

$(\tilde{\mathfrak{a}}, G,\tilde{\alpha})$

$\lambda$

$\tilde{u}(t)=u(t)\otimes\lambda(t)$

,

.

$t\in G$

Let be a compact neighbourhood of such that $p-\Omega\subset(Spue)^{c}$ and let be the spectral projection of corresponding to . Set $D=e_{1}Ke_{1}$ . Then is the identity of and and . We have that Sp since $0$

$\Omega$

$e_{1}$

$\ovalbox{\tt\small REJECT}^{Ad\lambda}(K)$

$ D\in$

$\Omega$

$\lambda$

$B\otimes D\in\ovalbox{\tt\small REJECT}^{\delta}(\tilde{\mathfrak{a}})$

$\overline{\tilde{\pi}(B\otimes D})^{w}$

$e\otimes e_{1}$

$\tilde{u}\cdot(e\otimes e_{1})Sp$

Sp $\tilde{u}\cdot(e\otimes e_{1})\subset Spue+Sp\lambda e_{1}=Spue+\Omega$ . , we can conclude that Sinoe Sp . Suppose . Then there are a covariant representation $(\pi, u)$ of and such that Sp $ue\ni O$ and Sp where is the identity of . $B\supset C^{*}(G)BC^{*}(G)$ , where First we assert that can be chosen to satisfy that $C^{*}(G)$ is identified with in . Set $\tilde{\Gamma}(\tilde{\alpha})Sp$

$\tilde{u}\cdot(e\otimes e_{1})\ni 0$

$p\not\in F(\tilde{\alpha})$


$ue\Rightarrow p$

$B\in\ovalbox{\tt\small REJECT}^{\partial}(\tilde{a})$

$\overline{\pi(B})^{w}$

$e$

$B$

$1\otimes C^{*}(G)$

$L=$

where $B$

.

{

$x\in\tilde{\mathfrak{a}}:\pi(x)u(f)=0;\forall f\in L^{1}(G)$

$u(f)=\int f(t)u(t)dt$

Since

$M(\tilde{\mathfrak{a}})$

,

supp $ f\cap Spue=\phi$ },

. Then is an -invariant closed left ideal of containing , we have that . Set . Then $L$

$\tilde{\alpha}$

$\tilde{\mathfrak{a}}$

$LC^{*}(G)\subset L$

$\overline{\pi}(C^{*}(G))\subset u(G)^{\prime}$

$B_{1}=L\cap L^{*}$

satisfies the above assertion. Now we assume that $B\supset C^{*}(G)BC^{*}(G)$ . Hence . Let be a compact neighbourhood of such that $p+\Omega-\Omega\subset(Spue)^{c}$ . Let $e(q)$ be the spectral , in particular projection of $\lambdarresponding$ to . Since the supremum of e$(q),$ is l, $thereisq_{0}\in\Gamma satisfyingSpue\overline{\pi}(e(q_{0}))\ni 0$ . Let and now we restrict the representation to with the representation space . $B_{1}\in\ovalbox{\tt\small REJECT}^{\partial}(\tilde{\mathfrak{a}})$

and

$B_{1}$

$e\in\overline{\pi}(C^{*}(G))^{\prime}$

$\Omega$

$0$

$ q-\Omega$

$q\in\Gamma,$

$e(q)\in M(\tilde{\mathfrak{a}})$

$inM(\tilde{\mathfrak{a}})$

$D=e(q_{0})Ke(q_{0})\in\ovalbox{\tt\small REJECT}^{Ad\lambda}(K)$

$\mathfrak{a}\otimes D\in\ovalbox{\tt\small REJECT}^{\partial}(\tilde{\mathfrak{a}})$

$ e(q_{0})\tilde{\mathfrak{a}}e(q_{0})\simeq$

$\mathcal{K}\equiv\overline{\pi}(e(q_{0}))\ovalbox{\tt\small REJECT}_{\pi}$

SIMPLE CROSSED PRODUCTS OF

$C^{*}$

-ALGEBRAS

We define the following unitary representation \û on

\û(t)=u Then \û commutes with

Sp

$\overline{\pi}(D)$

$(t)\overline{\pi}(\lambda(-t))\langle t, q_{0}\rangle$

, Ad

$\hat{\text{{\it \^{u}}}}e\cap\Omega\neq\phi$

,

$\mathcal{K}$

$t\in G$

for

$\hat{\text{{\it \^{u}}}}(t)(\overline{\pi}(a))=\overline{\pi}\circ\alpha_{t}(a)$

73

:

.

$a\in \mathfrak{a}$

, and

,

Sp $\theta e\subset Spue-Sp\lambda e(q_{0})+q_{0}\subset Spue+\Omega\subset(p+\Omega)^{c}$ .

Set $L=$

{

$x\in \mathfrak{a}\otimes D:$

\mbox{\boldmath $\pi$}(x)\û(f)=0,

$\forall f\in L^{1}$

supp

,

$ f\subset p+\Omega$

}

Then is an containing $e(q_{0})Be(q_{0})$ . .-invariant closed left ideal of Further $LD\subset L$ . Hence is of the form is an -invariant closed where left ideal of . Set , and let . Then Sp be the identity of , i.e. and Sp . Thus . for Now we consider the dual system and characterize $G(a)$ , similarly to Lemma 3.2 in [7]; $L$

$\alpha\otimes id$

$\mathfrak{a}\otimes D$

$L$

$L_{1}\otimes D$

$B_{1}=L_{1}\cap L_{1}^{*}$

$\mathfrak{a}$

$(p+\Omega)^{c}$

$L_{1}$

$\overline{\pi}(B_{1})$

$e_{1}$

$\text{{\it \^{u}}} e_{1}\cap\Omega\neq\phi$

$\alpha$

$ a_{e_{1}}\subset$

$p\not\in F(\alpha)$

$ q\in Sp\theta e_{1}\cap\Omega$

$p\not\in Sp\text{{\it \^{u}}} e_{1}-q$

$(\mathfrak{a}\times_{\alpha}G, \Gamma, \theta)$

Lemma 3.2. $G(\theta)=$

Proof. Let

$(\Omega_{\iota})$

{

$t\in G:\alpha_{t}(I)\subset I$

for any ideal

$I$

of }. $\mathfrak{a}$

Let be an ideal of and let $t\in G$ . Suppose that be a net of compact neighbourhoods of such that $I$

$\alpha_{t}(I)\zeta I$

$\mathfrak{a}$

$0$

$I_{\Omega}=\bigcap_{se\Omega}\alpha_{s}(I)$

We assert that elements and $e_{n}$

$\cap\Omega_{\iota}=(0)$

.

Let

.

is dense in . For let $x\in I$ be positive and find positive in the -subalgebra generated by such that $I$

$\bigcup_{\iota}I_{\Omega}$

$x_{n}$

.

$C^{*}$

$x$

$e_{n}x_{n}=x_{n}$

,

$\Vert x-x_{n}\Vert\leq 1/n$

.

Then by Lemma 3.2 in [7] there is . Thus such that for . Hence the closure of contains . Suppose that is also dense in for any . Since , a contradiction. Thus there is such that this would imply that $\alpha_{s}(x_{n})\in I$

$\Omega_{\iota}$

$I_{\Omega\iota}$

$ x_{n}\in$

$x$

$\bigcup_{\iota}I_{\Omega}$

$I_{\Omega-\Omega}\subset\alpha_{-t}(I_{\Omega})$

$\cup I_{\Omega-\Omega}$

$\ell$

$I\subset\alpha_{-t}(I)$

$\propto\alpha_{-t}(I_{\Omega})$

$s\in-\Omega_{\iota}$

$I$

,

$I_{\Omega-\Omega}$

$t$

.

Let contained in

$J=I_{\Omega_{\ell}-\Omega}$

$I_{\Omega}$

,

. Then since the ideal

$V_{s\in\Omega\iota}\alpha_{s}(J)$

we have that $J\not\subset_{s\in}v_{\Omega}\alpha_{s-\ell}(J)$

.

generated by

$\alpha_{s}(J)$

,

se

$\Omega_{\iota}$

is

74

AKITAKA KISHIMOTO

Let linear span of

and let

$B=\overline{J\cdot \mathfrak{a}\times_{\alpha}G\cdot J}\in\ovalbox{\tt\small REJECT}^{\hat{\alpha}}(\mathfrak{a}\times_{\alpha}G)$

$x\lambda(f)y$

,

$x,$

$y\in J$

,

$\Omega=\Omega_{\iota}$

.

Then

supp

with

$f\in L^{1}\cap L^{2}$

$B^{\theta}(t-\Omega)$

is the closed

.

$ f\subset t-\Omega$

Hence the $hereditary*$ -algebra generated by elements of the form is dense in $B(t-\Omega)$ . is -integrable, i.e. there is a positive $I(a)$ in (in fact in ) such that

$y^{*}\lambda(f)^{*}x^{*}x\lambda(f)y$

$a\equiv y^{*}\lambda(f)^{*}x^{*}x\lambda(f)y$

$\theta$

$M(\mathfrak{a}\times_{\alpha}G)$

$\mathfrak{a}\subset M(\mathfrak{a}\times_{\alpha}G)$

$\phi(I(a))=\int_{\Gamma}\phi(\theta_{p}(a))dp$

for every

$\phi\in(\mathfrak{a}\times_{\alpha}G)^{*}$

.

Explicitly

$ I(a)=\int|f(s)|^{2}y^{*}\alpha_{-s}(x^{*}x)yds\in J\cdot V\alpha_{s-t}(J)\equiv J_{1}s\in\Omega$

Since $B(t-\Omega)$ is -invariant, it follows that $I(a)\in B(t-\Omega)^{**}$ . Hence the hereditary -algebra generated by elements of the form $I(a)$ is contained in $B(t-\Omega)$ and of course is dense in $B(t-\Omega)$ . Hence $\theta$

$*$

$B(t-\Omega)\subset J_{1}\cdot \mathfrak{a}x_{\alpha}G\cdot J_{1}$

.

$B(t-\Omega)\neq B$ , i.e. Since . $G(a)$ Suppose that dg . Then there are a covariant representation $(\pi, u)$ of and such that Sp $ue\ni O$ and Sp $ue\geq t$ where is the identity of . Let be a compact neighbourhood of $O\in G$ such that . Then for any $x\in B$ and , $t\not\in G(\theta)$

$J_{1}\subset\neq J,$

$t$

$(\mathfrak{a}\times_{\alpha}G, \Gamma, \theta)$

$B\in\ovalbox{\tt\small REJECT}^{\theta}(\mathfrak{a}\times_{\alpha}G)$

$\overline{\pi(B)}^{w}$

$e$

$\Omega$

$ t+\Omega-\Omega$

$\subset(Spue)^{c}$

$ s\in\Omega$

$\pi(x)\overline{\pi}(\lambda(s))u(f)=0$

,

Let be the left ideal of with lation, the left ideal generated by $L$

with

$f\in L^{1}(\Gamma)$


$B=L\cap L^{*}$

$suppf\subset t+\Omega$

.

. Then from the

above calcu-

$L_{1}$

$\cup L\lambda(f),$

satisfies that for

$x\in L_{1}$

$f\in L^{1}(G)$

supp

with

$ f\subset\Omega$

,

,

$\pi(x)u(f)=0,$

Set and Sp . positive The cone of

$B_{1}=L_{1}\cap L_{1}^{*}\in\ovalbox{\tt\small REJECT}^{\theta}(\mathfrak{a}\times_{\alpha}G)$

$f\in L^{1}(\Gamma)$

and let

$e_{1}$

with supp

$ f\subset t+\Omega$

be the identity of

.

$\overline{\pi(B_{1})}^{w}$

$(*)$

.

Then Sp

$ue_{1}\ni 0$

$ue_{1}\ni t$

$a=\lambda(f)^{*}x^{*}x\lambda(f),$

$x\in B,$

ated by elements

$I(a)$

$B_{1}$

has a total set of -integrable elements of the form with supp . Let be the ideal of gener$\theta$

$f\in L^{1}\cap L^{2}$

with all such

$ f\subset\Omega$

$a$

.

Then

$J$

$B_{1}\subset\overline{J\cdot \mathfrak{a}\times_{\alpha}G\cdot J}$

$\mathfrak{a}$

. Sinoe

$\overline{J\cdot \mathfrak{a}\times_{l}G\cdot J}$


$C^{*}$

-ALGEBRAS

75

is generated by elements of the form $x_{1}I(a_{1})yI(a_{2})x_{2}$ with since $I(a_{1})yI(a_{2})\in B_{1}$ , we have that G. Set . Then satisfies $(*)$ sinoe mutes with . Henoe there is a compact neighbourhood of such that $x_{i}\in \mathfrak{a},$

$y\in \mathfrak{a}\times_{\alpha}G$

and

$J=\overline{JB_{1}J.}$

$\overline{J\cdot \mathfrak{a}\times_{\alpha}}$

$B_{2}=\overline{J\cdot \mathfrak{a}\times_{\alpha}G\cdot J}\in\ovalbox{\tt\small REJECT}^{\&}(\mathfrak{a}\times_{\alpha}G)$

$x\in B_{2}$

$\overline{\pi}(\mathfrak{a})$

$u(\Gamma)$

$\neq B_{2}$

$t$

$\Omega_{1}$

com-

$B_{2}(\Omega_{1})$

.

Since

is the closed linear span of

$B_{1}^{\text{{\it \^{a}}}}(\Omega_{1})$

$x\lambda(f)y,$

$x,$

$y\in J,$ $f\in L^{1}\cap L^{2}$

similarly to the first part of the proof, ated by elements of the form

$B_{2}(\Omega_{1})$

supp

with

,

$f\subset\Omega_{1}$

is the hereditary

$I(y^{*}\lambda(f)^{*}x^{*}x\lambda(f)y)=\int|f(s)|^{2}y^{*}\alpha_{-s}(x^{*}x)yds$

$C^{*}$

-subalgebra gener-

.

$q.e.d$ . Henoe , which implies that . G. Here we give a comment. Our referenoe on -integrability[2.4, 7] contains an error in the definition of . The correct form should be the one given in the above proof (otherwise Lemma 2.6 in [7] would fail), i.e. $a\in M(B)_{+}$ is -integrable if there is a (neoessarily unique) $I(a)\in M(B)$ such that $\alpha_{-t}(J)J$

$B_{2}(\Omega_{1})\supset\alpha_{-t}(J)J\cdot \mathfrak{a}\times_{a}$

$J\zeta\alpha_{-t}(J)$

$\theta$

$I$

$\beta$

$\phi(I(a))=\int_{\Gamma}\phi\circ\beta_{p}(a)dp$

, where for every is conand $\beta=a$ in this case. Here tinuous. To prove that the -integrable elements are hereditary, we adopt an argument similar to the one in [2.4, 7] by using, e.g., Lemma 2.1 in [1], although this fact is not quite necessary in the above proof, because we have considered only elements of (or ) as integrable elethe form if $a_{p}(y)=y,$ ments, which is justified by Proposition 2.8 in [7]. $\phi\in B^{*}$

$p\rightarrow\phi\circ\beta_{p}(a)$

$B=\mathfrak{a}\times_{\alpha}\backslash G$

$\beta$

$y^{*}\lambda(f)^{*}x^{*}x\lambda(f)y$

$\lambda(f)^{*}x^{*}x\lambda(f)$

Lemma 3.3 [10]. The isomorphic to .

$C^{*}$

$ p\in\Gamma$

-dynamical system

$(\mathfrak{a}\times_{\alpha}G\times a^{\Gamma}’ G,\hat{\theta})$

is covariantly


Lemma 3.4. $F(\alpha)=$

{

$p\in\Gamma:\theta_{p}(I)\subset I$

for any ideal

$I$

of


}

Proof. $ItfollowsfromLemmas3.2and3.3$ . By the above lemma and Lemma 3.1 in [7] we have Theorem3.5.

lent:

Let


be as above.

Thefollowing conditions are equiva-

AKITAKA KISHIMOTO

76

is simple; is -simple and . As a corollary to Lemma 3.4 we give

(i) (ii)


$\mathfrak{a}$

$ F(\alpha)=\Gamma$

$\alpha$

Proposition 3.6. Let be as above. Suppose that there is another $C^{*}- dynamic\dot{a}l$ system $(B, G, \beta)$ which is exteriorly equivalent to $(a, G, \alpha)$ . Then $(\mathfrak{a}, G, \alpha)$

.

$F(\alpha)=F(\beta)$

Proof.

$(\mathfrak{a}\times_{\alpha}G, \Gamma, a)$

is covariantly isomorphic to

$(B\times\beta G, \Gamma,\hat{\beta})$

.

It seems more difficult to compute than in most of cases. times coincides with . We shall show some of these cases. The following lemma can be found, e.g. in [11, Lemma 22]: $F(\alpha)$

$F(\alpha)$

$\Gamma(\alpha)$

$q.e.d$

.

But some-

$\Gamma(\alpha)$

Lemma 3.7. Suppose that primitive ideal I of , where

is

$a$

$\alpha$

-simple and that

$G/G_{I}$

is compact

for any

$\mathfrak{a}$

$G_{I}=\{t\in G:\alpha_{t}(I)=I\}$

.

Then the primitive ideal space of with the transposed action of is isomorphic to $G/G_{0}$ with the action of by translations, where $G_{0}=G_{I}$ for any primitive ideal . $G$

$\mathfrak{a}$

$G$

$I$

Proof. Let be a primitive ideal, and let hoods of $0\in G/G_{I}$ such that . Sinoe same image in $G/G_{I}$ , we can define $I$

be a net of compact neighbourif and in have the

$(\Omega_{\iota})$


$\alpha_{t}(I)=\alpha_{s}(I)$

$t$

$G$

$s$

$f=\dot{s}$

.

$I(f+\Omega_{\iota})=\bigcap_{S\in\Omega}\alpha_{\ell+s}(I)$

There is a finite set

$S_{\iota}$

of

$G/G_{I}$

such that

$\bigcup_{S\in S}(\dot{s}+\Omega_{\iota})=G/G_{I}$

Sinoe

$\mathfrak{a}$

is -simple, we have $\alpha$

$\bigcap_{\dot{s}\in S}I(\dot{s}+\Omega_{\iota})=(0)$

Let

.

$J$

be a primitive ideal.

Sinoe

.

$\bigcap_{\dot{s}\in S_{\iota}}I(\dot{s}+\Omega_{\iota})\subset J$

, there is

an

$\dot{s}_{\iota}\in S_{\iota}$

such

that $I(\dot{s}_{\iota}+\Omega_{\iota})\subset J$

Sinoe $\dot{s}+\Omega_{\iota}\ni\dot{s}_{j}$

.

is compact, we may suppose that for sufficiently large . Henoe $G/G_{I}$

$ j\geq\ell$

$\dot{s}_{\iota}$

converges, say to . Then which implies

$\dot{s}+\Omega_{\iota}+\Omega_{\iota}\supset\dot{s}_{j}+\Omega_{j}$

$I(\dot{s}+\Omega.+\Omega.)\subset J$

.

$\dot{s}$


$C^{*}$

-ALGEBRAS

77

We have that , sinoe , as shown from is dense in the first part of the proof of Lemma 3.2 and the fact that the quotient map is open. Similarly we get that $\alpha_{t}(J)\subset Iforsomet\in G$ , i.e. $\alpha_{s}(I)\subset J$

$\cup I(\dot{s}+\Omega+\Omega_{\iota})$

$\alpha_{s}(I)$

$G\rightarrow G/G_{I}$

$\alpha_{s+t}(J)\subset\alpha_{s}(I)\subset J$

.

Sinoe $G/G_{J}$ is compact, we can conclude that $\alpha_{s+c}(J)=J$ . Thus . Hence the set of primitive ideals is , i.e. there is a correspondenoe between the primitive ideal space and $G/G_{I}$ , by choosing one primitive ideal of , which obviously preserves the actions of . For any subset of $G/G_{I}$ , we have $\alpha_{s}(I)=J$

$one- 0\acute{n}e$

$\{\alpha_{t}(I):t\in G/G_{I}\}$

$I$

$G$

$\mathfrak{a}$

$\bigcap_{\dot{s}\in S}\alpha_{s}(I)=\bigcap_{\dot{s}\in S}\alpha_{s}(I)$

$S$

.

By the same argument as above, $\bigcap_{l\in S}\alpha_{s}(I)\subset\alpha_{t}(I)$

implies that

$i\in\overline{S}$

.

Henoe the closure operations coincide through the correspond-

ence.

$q.e.d$

Proposition 3.8.

Let $ abeliangroupand\mathfrak{a}is\alpha$ -simple.


be a Then

$C^{*}$

-dynamical system where

$\Gamma(\alpha)=F(\alpha)$

$G$

.

is a discrete

.

Proof. We apply Lemma 3.7 to the dual system , where now is compact, and use the formula for in Lemma 3.4 and the one for in Corollary 5.4 in [7].

$\Gamma$

$(\mathfrak{a}\times_{\alpha}c, \tau, a)$

$ff(\alpha)$

$\Gamma(\alpha)$

Proposition 3.9. Let be a separable -dynamical system (i.e. both and are separable). Suppose that is discrete for any primitive ideal I of a where $(a, G, \alpha)$

$G$

$a$

$C^{*}$

$G_{l}$

$G_{I}=\{t\in G:\alpha_{t}(I)=I\}$

If is of of

.

-simple, then the primitive ideal space of with the transposed action is isomorphic to (with action of ) for some closed subgroup $H$ of $G$ such that is compact, in particular $F(\alpha)=\Gamma(\alpha)(=H)$ . $a$

$\alpha$

$a$


$\Gamma$

$\Gamma/H$

$\Gamma$

$\Gamma/H$

Proof. (c.f. [8, Theorem 3.1]) By [5, Corollary 3.2], for any primitive ideal is compact. Apply Lemma 3.7. For the last statement, see the proof of Proposition 3.8. For applications of the above proposition we refer the reader to [8]. $I,$

$\Gamma/\Gamma_{I}$

AKITAKA KISHIMOTO

78

\S 4. Crossed products of

$O_{n}$

Let be an n-dimensional Hilbert spaoe $(2\leq n\leq\infty)$ , and let be the $O(f)$ . For each Fock Hilbert spaoe over is a bounded operator defined by $F(\ovalbox{\tt\small REJECT}_{n})$

$\ovalbox{\tt\small REJECT}_{n}$

$f\in\ovalbox{\tt\small REJECT}_{n},$


$O(f)g_{1}\otimes\cdots\otimes g_{n}=f\otimes g_{I}\otimes\cdots\otimes g_{n}$

$O(f)\Omega=f$

$O(f)$ is an isometry. . If where is the vacuum vector in We de-algebra generated by $O(f),$ . the note by If is finite, the Cuntz algebra $O_{n}[3]$ is isomorphic to the quotient of by the compact operator algebra on and if $n=\infty,$ . is isomorphic to induces an automorphism of through that of Each unitary on defined by $\Omega$

$\Vert f\Vert=1,$

$F(\ovalbox{\tt\small REJECT}_{n})$

$C^{*}$

$O(\ovalbox{\tt\small REJECT}_{n})$

$f\in\ovalbox{\tt\small REJECT}_{n}$

$n$


$O_{n}$

$F(\ovalbox{\tt\small REJECT}_{n})$

$u$


$O_{n}$


$O(f)\rightarrow 0(uf)$

,

$f\in\ovalbox{\tt\small REJECT}_{n}$


.

We call quasi-free those automorphisms of obtained in this way. See, for the detail, Evanoe [4]. From now on we assume that is finite. Let be a locally compact abelian group with its dual , as before, and let be a continuous unitary representation of $G$ on . By thinking of elements of the form $O_{n}$

$n$

$G$

$\Gamma$

$u$


$O(f_{1})\cdots O(f_{n})O(g_{1})^{*}\cdots O(g_{m})^{*}$

it is clear that Sp is the closed subgroup of $\alpha$

Lemma 4.1. Proof. $withp_{i}\in\Gamma$

Let

.

Let

$(O_{n}, G, \alpha)$

$(\phi_{i})_{i=1}^{n}$

be as above.

$\Gamma$

generated by Sp

Then

$\Gamma(\alpha)=Sp\alpha$

be an orthonormal system in

$LetS_{\ell}betheimageofO(\phi_{i})intoO_{n}$

Set for each

,

$X_{n}$

$u$

.

.

such that

$u_{t}\phi_{i}=\langle t, p_{i}\rangle\phi_{i}$

.

$ k=1,2,\ldots$

$S_{i}^{1k)}=\sum_{\mu:\ell\langle\mu)=k}S_{\mu}S_{i}S_{\mu}^{*}$

,

where the summation is taken over all the words of 1, , with length $l(\mu)=k$ , (c.f. [3]). All are isometries and satisfy that , and $x]\Vert=0$ for any is the algebra of fixed points under the gauge where ; automorphism group , i.e. the quasi-free automorphism group induced by { $\mu$

$S_{i^{k)}}^{t}$

$S_{\{\ell_{1},\ldots.i_{k}\}}=S_{i_{1}}\cdots S_{i_{k}}$

$x\in F^{n}\subset O_{n}$

$\gamma$

$|z|=1\}$

on


.

$\{$

$\ldots$

$n\}$

$\lim\Vert[S_{\ell}^{tk)}$

$F^{n}$

$z\cdot 1$


$C^{*}$

-ALGEBRAS

79

be an infinite aperiodic sequenoe of letters 1, , (c.f. Lemma 1.8 in [3]) $andletv_{m}$ be the restriction ofv to the firstm letters. Set

Let

$v$

$\ldots$

$Q_{m}=\sum_{\mu,l\langle\mu)=m}S_{\mu}S_{v_{m}}S_{v_{m}}^{*}St$

Then

$n\}$

$\{$

$\{Q_{m}\}$

are

$\gamma-$

.

and -invariant projections and satisfy $\alpha$

$\lim\Vert Q_{m}\epsilon(x)Q_{m}-Q_{m}xQ_{m}\Vert=0$ $\lim$

I

$ Q_{m}xQ_{m}\Vert=\Vert\epsilon(x)\Vert$

where is the projection of Then for any positive , $\epsilon(x)=\int\gamma_{t}(x)dt$

$0_{n}$

onto

$F^{n}$

.

Proposition 1.7 in [3]).

( $c.f$

$x\in O_{n}$

$\lim_{m}\lim_{k}\Vert Q_{m}xQ_{m}S_{i}^{1k)}x\Vert=\lim$ $\lim\Vert S_{i}^{(k)}Q_{m}\epsilon(x)Q_{m}x\Vert$

.

$=\lim_{m}\Vert Q_{m}\epsilon(x)Q_{m}x\Vert\geq\Vert\epsilon(x)\Vert^{2}$

Let zero positive

and with

$B\in\ovalbox{\tt\small REJECT}^{\alpha}(O_{n})$

$x\in B$

$\Omega$

a compact neighbourhood of

$Sp_{\alpha}(x)\subset\Omega$

.

$aremandksuchthatxQ_{m}S_{i}^{(k)}x\neq 0$

.

.

Then there is a nonIt follows from the above calculation that there This implies that $0$

Sp $\alpha|B\cap(p_{i}+\Omega+\Omega)\neq\phi$

Sinoe and are arbitrary, we can conclude that . Thus sinoe is a closed subgroup [6]. We denote by the extension of the gauge action to an action on . This is possible because commutes with . In the following we denote by $H$ the intersection of the closed subsemigroups of generated by Sp and $-p$ , with $B$

$\Omega$

$\Gamma(\alpha)=Sp\alpha$

$\Gamma(\alpha)\ni p_{i}$

$\Gamma(\alpha)$

$\overline{\gamma}$

$O_{n}\times_{\alpha}G$

$\gamma$

$\alpha$

$\gamma$

$\Gamma$

$p\in Spu$

.

Lemma4.2. Let


Proof.

be as above.

We construct certain $ k=1,2,\ldots$ , set

$\alpha$

.

$ThenH\supset F(\alpha)$

-invariant states of

$P_{i}^{(k)}=S_{\ell}^{k}S_{i}^{*k}$

$O_{n}$

.

For

$i=1,\ldots,$

$\{P_{i}^{1k)}\}_{k=1,2},\ldots$

$\gamma$

$\phi_{i}(x)=\phi_{\ell}(P_{i}^{\langle k)}xP_{\ell}^{(k)})$

Then it is shown that

$\phi_{i}$

,

$x\in O_{n}$

$n$

and

.

is a decreasing sequenoe of -invariant projections. -invariant state satisfying

Then $\gamma$

$u$

,

$ k=1,2,\ldots$

Let

$\phi_{\ell}$

be a

.

is -invariant (and in fact unique) and that the continuous $\alpha$

AKITAKA KISHIMOTO

80

functions on ated by characters

for and

$G\ni t\rightarrow\phi(x\alpha_{t}(y))$

$p_{1},\ldots,$

$p_{n}$

$x,$

$y\in O_{n}$

$-p_{\ell}$

are contained in the closed algebra gener-

, e.g.

$\phi_{i}(x\alpha_{t}(S_{\mu}S_{v}^{*}))=\langle t, p_{j_{1}}+\cdots+p_{j_{m}}-p_{k_{1}}-\cdots-p_{k_{n}}\rangle\phi_{i}(xS_{\mu}S_{v}^{*})$

, where . This implies is non-zero only if that in the GNS representation associated with , the canonical representation $U$ of defined by $\mu=\{j_{1},\ldots, j_{m}\}$

$v\equiv\{k_{1},\ldots, k_{n}\}=\{i,\ldots, i\}$

$\phi_{i}$

$G$

$U_{t}\pi_{\phi_{i}}(x)\Omega_{\phi_{\ell}}=\pi_{\phi_{i}}\circ\alpha_{t}(x)\Omega_{\phi_{i}}$

has spectrum in the closed subsemigroup . Proposition 2.2 we have that

$H_{i}$

,

$x\in O_{n}$

generated by

$p_{1},\ldots,$

$p_{m}$

and

$-p_{i}$

. By

$\tilde{\Gamma}(\alpha)\subset H_{l}$

Lemma 4.3. Let , it follows that



Proof.

Let

$a_{p}(I)\subset I$

for

$p\in H$

$\overline{\gamma}$

be a representation of or be of the form , we have


$\rho$

of

.

$x\in O_{n}^{\gamma}\times_{\alpha}G\equiv(O_{n}\times_{\alpha}G)^{\overline{\gamma}}$

sinoe

Then for any -invariant ideal I

be as above.

whose kemel is -invariant. Let with . Then $\overline{\gamma}$

$\sum a_{\iota}\otimes f_{i}$

$a_{i}\in O_{n}^{\gamma},$

$f_{i}\in C^{*}(G)$

$\lim\Vert S_{\ell}^{(k)*}xS_{i}^{(k)}-\theta_{p\ell}(x)\Vert=0$

$\lim_{k}\Vert\rho(S_{i}^{(t)*}xS_{i}^{(k)})\Vert=\Vert\rho\circ\theta_{p\ell}(x)\Vert$

.

The left hand side equals $\lim_{k}\Vert\rho(x)\rho(S_{i}^{1k)}S_{i}^{1k)*})\Vert$

sinoe tions and

$S_{i}^{(k)}S_{i}^{(k)*}$

asymptotically commutes with , we have

$x$

.

Further sinoe

$S_{i}^{(k)}S_{i}^{(k)*}$

are projec-

$\sum_{i}S_{t^{k)}}^{t}S^{(k)*}=1$

$|1\rho(x)\Vert\geq\Vert\rho\circ a_{p\iota}(x)\Vert$

$(*)$

$\Vert\rho(x)\Vert=\max_{i}\Vert\rho\circ\theta_{p\ell}(x)\Vert$

For a fixed

$x\in O_{n}^{\gamma}\times_{\alpha}G$

we can find an infinite sequence

$\{i_{k}\}$

of 1, $\{$

$\ldots$

,

$n\}$

such

that $(**)$

$\Vert\rho(x)\Vert=\Vert\rho\circ a_{p\iota_{1}+\cdots+p\ell_{k}}(x)\Vert$

for all $\{i_{k}\}$

.

.

There is an For such an we have $ k=1,2,\ldots$

$i\in\{1,2,\ldots, n\}$

which infinitely often appears in

$i$

$\Vert\rho(x)\Vert=\Vert\rho\circ a_{np\ell}(x)\Vert$

for

$ n=1,2,\ldots$

, sinoe for any subset

$J$

of 1, $\{$

$\ldots$

,

$k\},$

$(**)$

is less than or equal to


$C^{*}$

$\Vert\rho\circ\theta_{(\sum_{j\in J^{p\ell_{j}}})}(x)\Vert\leq\Vert\rho(x)\Vert$

Let ated by

$p\in H$

-ALGEBRAS

81

.

. By the assumption there is a sequenoe in the subsemigroup generwhich converges to , i.e. there is a sequenoe

$\{p_{1},\ldots, p_{n}, -p_{i}\}$

$p$

$q_{l}=\sum_{k}n_{k}^{1l)}p_{k}-m^{(l)}p_{i}-p,$

which converges to zero in

$\Gamma$

.

$n_{k}^{(l)}\geq 0,$

$m^{(\ell)}\geq 0$

Then

$\Vert x-a_{q_{i}}(x)\Vert\geq\Vert\rho\circ\theta_{m^{(i)}p\ell}(x)-\rho\circ\alpha_{(\Sigma n_{k}^{(l)}p\kappa^{-p)}}(x)\Vert$

$\geq\Vert\rho(x)\Vert-\Vert\rho\circ\theta_{-p}(x)\Vert$

which implies that

$\Vert\rho\circ a_{-p}(x)\Vert\geq\Vert\rho(x)\Vert$

. Henoe

$\theta_{p}(I)\cap O_{n}^{\gamma}\times_{\alpha}G\subseteq I\cap O_{n}^{\gamma}\times_{\alpha}G$

where that

$I$

is the kernel of

$a_{p}(I)\subset I$

$\rho$

.

Since

$I$

is generated by

$I\cap O_{n}^{\gamma}\times_{a}G$

,

we can conclude

.

$q.e.d$

Theorem 4.4. Let be as above. The crossed product simple if and only if the closed subsemigroup of generated by Sp and itselffor any $p\in Spu$ . $(O_{n}, G, \alpha)$


$\Gamma$

Proof.

Sinoe

$H\supset 5(\alpha)byLemma4.2,$

$ifH\neq\Gamma,$

$u$

$O_{n}\times_{a}$

$-p$

is

.

is $\Gamma$

Gisnot simple by Theo-

rem 3.5.

a-

Suppose $ H=\Gamma$ . Then Lemma 4.3 implies that any -invariant ideal is invariant. Sinoe is simple [3], there are not any non-trivial -invariant ideals of . Thus is -simple. Sinoe is prime by Lemma 4.1 and [7, Theorem 5.8], it follows from [7, Lemma 6.4] (or Lemma 3.7) that is simple. $\overline{\gamma}$

$O_{n}$



$\theta$


$\overline{\gamma}$

$O_{n}\times_{a}G$

Proposition 4.5. Let be as above and suppose that contains the gauge automorphism group . Then is the intersection of the closed subsemigroups of generated by Sp and $-p$ , with $p\in Spu$ . $(O_{n}, G, \alpha)$

$\alpha(G)$


$\gamma$

$\Gamma$

Sinoe is inner under the above assumption, any ideal of -invariant. The rest of the proof follows from Lemmas 4.2, 4.3 and 3.4. Proof.

$\overline{\gamma}$

$u$

$\overline{\gamma}$

\S 5. Crossed products of


is

$q.e.d$

.

$O_{\infty}$

In this section we consider the case $ n=\infty$ . As in Section 4, let be a weakly continuous unitary representation of a locally compact group $G$ on a separable infinite-dimensional Hilbert spaoe and let be the corresponding quasi-free $u$

$\ovalbox{\tt\small REJECT}$

$\alpha$

AKITAKA KISHIMOTO

82

action on by Sp . Let

$O_{\infty}=O(\ovalbox{\tt\small REJECT})$

.

It is immediate that Sp

$\alpha$

is the closed subgroup generated

$u$

$O_{\infty}$

on

be the Fock spaoe and the Fock representation of be the canonical representation of on the Fock spaoe, i.e.

$F=F(\ovalbox{\tt\small REJECT})=\sum_{0}^{\infty\ovalbox{\tt\small REJECT}\otimes n}$

$F[4]$

.

Let

$U_{F}$

$\pi_{F}$

$G$

$U_{F}(t)|\ovalbox{\tt\small REJECT}\otimes n=u_{t}\otimes\cdots\otimes u_{t}(n- tuples)\equiv u_{t}^{\otimes n}$

.

is the closed subsemigroup $H$ generated by Sp . The It is clear that Sp of in an obvious way. If gives a representation pair is not simple (c.f. [9]). is not faithful, in particular $U_{F}$

$u$

$(\pi_{F}, U_{F})$

$H\neq\Gamma,$

$\pi_{F}\times U_{F}$

$O_{\infty}\times_{\alpha}G$

$0_{\infty}\times_{\alpha}G$


be as above. The crossed product Theorem 5.1. Let simple if and only if the closed subsemigroup $H$ generated by Sp is $(0_{\infty}, G, \alpha)$

$\Gamma$

$u$


is

.

, then is not simple. Henoe we Proof. We have shown that if $ H=\Gamma$ . now assume that is faithful. Sinoe is irreducible, First we want to show that is prime. this in particular implies that onto For each $ n=1,2,\ldots$ , there is a natural unitary map $W$ from , such that where with and if . Note that $ H\neq\Gamma$





$F\otimes\ovalbox{\tt\small REJECT}^{\otimes n}$

$ W_{n}(\psi\otimes\phi)=\sum$

$\sum_{n}^{\infty}\ovalbox{\tt\small REJECT}^{\otimes k}cF$

$\psi_{k}\otimes\phi$

$\psi=\sum_{0}^{\infty}\psi_{k}$

$\psi_{k}\in\ovalbox{\tt\small REJECT}\otimes k$

$\psi_{k}\otimes\phi\in\ovalbox{\tt\small REJECT}^{\otimes k+n}$

$W_{n}\cdot U_{F}(t)\otimes u_{t}^{\otimes n}=U_{F}(t)W_{n}$

.

In the following, however, we omit $W$ . be a decreasing sequenoe of compact neighbourhoods of in such. that Let Sp and with such that . There are tends to zero weakly. and due to the assumption, there are positive . Then, sinoe Sp Let cp-p, with such that Sp . integers $m$ and where are monomials (i.e. of the type Let $0$

$(\Omega_{\iota})$


$u$

$ p_{\iota}\in$

$\phi_{\iota}\in\ovalbox{\tt\small REJECT}$

$\Vert\phi_{\iota}\Vert=1$

$\Gamma$

$Sp_{u}\phi_{\iota}\subset p_{\iota}+\Omega_{\iota}$

$\phi_{\iota}$

$ U_{F}=\Gamma$

$ p\in\Gamma$

.

$\Vert\psi_{\iota}\Vert=1$

$\psi_{\iota}\in\ovalbox{\tt\small REJECT}\otimes m$

$x=\sum a_{k}\otimes f_{k}\in O_{\infty}\times_{\alpha}G$

and Note that for any

$O(f_{\ell})O^{*}(g_{1})\cdots O^{*}(g_{j}))$

$f_{k}\in L^{1}(G)$

$\psi\in F$

$U_{F}\psi_{\iota}$

$+\Omega_{\iota}$

$ O(f_{1})\cdots$

$a_{k}$

.

,

$\lim\Vert\int f_{k}(t)U_{F}(t)(\psi\otimes\phi_{\iota}\otimes\psi_{\iota})dt-\int f_{k}(t)\langle t, p\rangle(U_{F}(t)\psi)\otimes\phi_{\iota}\otimes\psi_{\iota}dt\Vert=0$

$\lim\Vert\pi_{F}(a_{k})(\psi\otimes\phi_{\iota}\otimes\psi_{\iota})-(\pi_{F}(a_{k})\psi)\otimes\phi_{\iota}\otimes\psi_{\iota}\Vert=0$

.

Henoe we have that $\lim\Vert(\pi_{F}\times U_{F})(x)(\psi\otimes\phi_{\iota}\otimes\psi_{\iota})-((\pi_{F}xU_{F})\circ(\theta_{p}(x)\psi))\otimes\phi_{\iota}\otimes\psi_{\iota}\Vert=0$

Since

$\psi\in F$

is arbitrary, we have that

$\Vert\pi_{F}\times U_{F}(x)\Vert\geqq\Vert(\pi_{F}\times U_{F})\circ\theta_{p}(x)\Vert$

.

, which in


turn implies, since

$C^{*}$

-ALGEBRAS

83

is arbitrary,

$p$

$\Vert(\pi_{F}\times U_{F})\circ a_{p}(x)\Vert=\Vert(\pi_{F}\times U_{F})(x)\Vert$

,

$ p\in\Gamma$

.

is simple [3], we can conclude that Sinoe is faithful. Now let be any irreducible representation of . By a similar reason as given in the proof of Theorem 4.4 it suffioes to show that ker $O_{\infty}$


$\pi\times U$


$(\pi\times U)\cap$

$O_{\infty}^{\gamma}\times_{\alpha}G=(0)$

.

Let be a $mplete$ orthonormal system of M. Then mutually orthogonal projections. Let $(f_{i})$

$P=\sum_{i=1}^{\infty}\pi(O(f_{\ell})O^{*}(f_{i}))$

$\pi(O(f_{i})O^{*}(f_{\ell}))$

are

.

. which is independent from choice of $P\neq 1$ . Let be a projection in such that $0\neq e\leqq 1-P$ . Then by the irreducibility, we have that . But , i.e. $1-P=e$ . Hence $1-P$ is one-dimensional. Now it is easily shown that is equivalent to and that $U=pU_{F}$ with some . Hence the faithfulness of follows from the above. Suppose that $P=1$ . Let $p\in Spu+\cdots+Spu$ ( terms). There is a sequence of unit vectors with such that Sp given before. Now we define a family of isometries:

$P$

is a projection in First suppose that

$U(G)^{\prime}$

$(f_{\ell})$

$U(G)^{\prime}$

$e$

$(1-P)[\pi(O_{\infty})e\ovalbox{\tt\small REJECT}_{n}]$

$[\pi(O_{\infty})e\ovalbox{\tt\small REJECT}_{\pi}]=\ovalbox{\tt\small REJECT}_{\pi}$

$=e\ovalbox{\tt\small REJECT}_{\pi}$

$\pi$

$ p\in\Gamma$

$\pi_{F}$

$\pi\times U\simeq\pi_{F}\times u_{F}\circ a_{p}$

$k$

$\psi_{1\iota}\otimes\cdots\otimes\psi_{k\iota}\in\ovalbox{\tt\small REJECT}\otimes k$

$U_{F}(\psi_{1\iota}\otimes\cdots\otimes\psi_{k\iota})cp+\Omega_{\iota}$

$\Omega_{\iota}$

$V_{\iota}^{t0)}=\pi(O(\psi_{1\iota})\cdots O(\psi_{k\iota}))$

$V_{\iota}^{tm})=\sum\pi(O(f_{i}))V^{(m-1)}\pi(O^{*}(f_{\ell}))$

Note that Let

$V|^{m}$

)

.

does not depend on . where are monomials, say, . For we have that $(f_{i})$

$x=\sum a_{k}\otimes f_{k}\in O_{\infty}^{\gamma}\times_{\alpha}G$

$a_{k}$

$a_{k}=O(g_{1})\cdots O(g_{m_{k}})$

.

$m\geqq m_{k}$

$O^{*}(h_{1})\cdots O^{*}(h_{m_{k}})$

$\pi(a_{k})V_{\iota}^{(m)}=V_{\iota}^{(m)}\pi(a_{k})$

$\lim\Vert U_{t}V_{\iota}^{(m)}-\langle t, p\rangle V_{l}^{(m)}U_{t}\Vert=0$

Thus, for

.

$m\geq\max(m_{k})$

$\lim\Vert(\pi\times U)(x)V^{(m)}-V_{l}^{(m})(\pi\times U)\circ\theta_{p}(x)\Vert=0$

) Sinoe are isometries, we have that is dense in , we have $V_{\iota}^{1m}$

$p$

.

$\Vert(\pi\times U)(x)\Vert\geq\Vert(\pi\times U)\circ\theta_{p}(x)\Vert$

$\Gamma$

$\Vert(\pi\times U)(x)\Vert=\Vert(\pi\times U)\circ\theta_{p}(x)\Vert$

,

$\forall p\in\Gamma$

.

.

Sinoe such

AKITAKA KISHIMOTO

84

Sinoe

$0_{\infty}$

is simple, we know that

$(\pi\times U)|O_{\infty}^{\gamma}\times_{\alpha}G$

is faithful.

added by those of the essential Remark 5.2. If the set of elements of Sp contains the gauge automorphism itself and if spectrum of is equal to Sp equals Sp . For example group , then it follows from the above proof that $U_{F}$

$\alpha(G)$

$U_{F}$

$u$


$U_{F}$

$\gamma$

$r_{(\gamma)=Z_{+}}$

.

of In passing we give a remark on a quasi-free automorphism tends to zero weakly as such that induoed by a unitary on following proposition implies, in particular, that the Fock state is the only . state of

$O_{\infty}$

$\alpha_{u}$

$u^{n}$

$\ovalbox{\tt\small REJECT}$

$u$

which is . The

$ n\rightarrow\infty$

$\alpha_{u}$

-invariant

$0_{\infty}$

Proposition5.3.

$Let\alpha_{u}$

be as above.

$Thenforanyx\in O_{\infty}$

,

$M_{N}(x)\equiv(2N+1)^{-1}\sum_{n=-N}^{N}\alpha_{u}^{n}(x)$

converges in norm to a multiple

of the

identity.

Suppose $x=O(f_{1})\cdots O(f_{n})O^{*}(g_{1})\cdots O(g_{m})^{*}$ with $=\cdots=\Vert g_{m}\Vert=1$ . Then if $n\geq 1$ ,

Proof.

$n+m\geq 1$ ,

and

$\Vert f_{1}\Vert$

$\Vert M_{N}(x)\Vert^{2}=\Vert M_{N}(x)^{*}M_{N}(x)\Vert$

$\leq(2N+1)^{-}:_{m=-N}\sum^{N}|\langle u^{n}f_{1}u^{m}f_{1}\rangle|$

. which implies that $\lim\Vert M_{N}(x)\Vert^{2}=0$ . Similarly we have the same in case , which is dense in The linear span of 1 and elements of the fom completes the proof. is with With alittle more care we can conclude that the system weakly asymptotically abelian. $m\geqq 1$

$O(f_{1})\cdots O^{*}(g_{m})$

$(0_{\infty}, Z, \alpha)$

$O_{\infty}$

$\alpha_{n}\equiv\alpha_{u^{n}}$

Note and Acknowledgements

The problems treated in this note were brought up by H. Takai to a series of seminars, held in Tokyo, attended by A. Ikunishi and Y. Nakagami besides us. The basic ideas of Section 5 were obtained during the seminars, and those of Sections 2 and 3 are a result of many discussions with H. Takai during the summer in 1979. I would like to thank the other members of the seminars and especially Dr. H. Takai for stimulating discussions. I am grateful to Professor D. W. Robinson for warm hospitality at the University of New South Wales, Sydney, Australia, where this was completed. I am also grateful to Dr. D. Olesen for a couple of comments.


$C^{*}$

-ALGEBRAS

85

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Olesen, D., Pedersen, G. K. : Applications of the Connes spectrum to -dynamical systems. J. Functional Analysis 30, 179-197 (1978). Olesen, D., Pedersen, G. K. : APplications of the Connes sPectrum to -dynamical systems II (preprint). Pedersen, G. K., Takai, H. : Crossed products of -algebras by aPproximately uniformly continuous actions, Math. Scand. 45, 282-288 (1979). Takai, H. : On a duality for crossed products of -algebras, J. Functional Analysis 19, $C^{*}$

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Department of Mathematics Yokohama City University

Yokohama, Japan