Strong topological transitivityand $C^{*}$ -dynamical ... - Project Euclid

J. Math. Soc. Japan Vol. 37, No. 1, 1985

Strong topological transitivity and

$C^{*}$

-dynamical systems

By Ola BRATTELI, George A. ELLIOTT

and Derek W. ROBINsON (Received Feb. 13, 1984)

$0$

.

Introduction.

denote the action of a group $H$ as homeomorphisms of a topological space $X$ ; then (X, $H,$ $T$ ) is said to be topologically transitive if for each pair of (see, non-empty open sets $A,$ there exists an $h\in H$ such that -dynamical sysfor example, [13] Chapter 5). Following [10], we define the to be topologically transitive if for each pair of non-zero elements tem . there exists an $h\in H$ such that is abelian. In this case The algebraic definition is particularly natural if $H$ of as homeomorphisms of the spectrum $X$ of such determines an action , for , and is topologically transiand that tive if and only if (X, $H,$ ) is topologically transitive. In [10] the definition is given in a slightly different form. These authors require that the product of each pair of non-zero -invariant hereditary -subalgebras is non-zero. This obviously follows from our definition such that $x\tau_{h}(y)=0$ for all $h\in H$ but conversely if there exist non-zero -subalgebras of the -invariant hereditary and then the product must be zero. and generated by Although the foregoing definition of transitivity is quite natural there is a -dynamical seemingly stronger notion which appears to be more useful. The topologically strongly is defined to be transitive if for each system for which finite sequence $\{(x_{i}, y_{i});i=1,2, \cdots , n\}$ of pairs of elements

Let

$T$

$B\subseteqq X$

$A\cap T_{h}(B)\neq\emptyset$ $C^{*}$

$(\mathcal{A}, H, \tau)$

$x,$

$x\tau_{h}(y)\neq 0$

$y\in \mathcal{A}$

$\mathcal{A}$

$\tau’$

$\mathcal{A}$

$\tau$

$(\tau_{h}x)(\omega)=x(\tau_{h-1}’\omega)$

$\omega\in X$

$x\in \mathcal{A}$


$\tau’$

$\mathcal{A}_{1}\mathcal{A}_{2}$

$C^{*}$

$\mathcal{A}_{1},$

$\tau$

$\mathcal{A}_{2}\subseteqq \mathcal{A}$

$x,$

$\mathcal{A}_{1}\mathcal{A}_{2}$

$y\in \mathcal{A}$

$C^{*}$

$\tau$

$\{\tau_{h}(x)^{*}\mathcal{A}\tau_{h}(x);h\in H\}$

$\mathcal{A}_{1}$

$\mathcal{A}_{2}$

$\{\tau_{h}(y)\mathcal{A}\tau_{h}(y)^{*} ; h\in H\}$

$C^{*}$


$x_{i},$

$\sum_{i=1}^{n}x_{\ell}\otimes y_{i}\neq 0$

in the algebraic tensor product

$\mathcal{A}\otimes \mathcal{A}$

$y_{i}\in \mathcal{A}$

,

, there exists

an

$h\in H$

such that

$\sum_{i=1}^{n}x_{i}\tau_{h}(y_{i})\neq 0$

in

$\mathcal{A}$

.

Clearly strong topological transitivity implies topological transitivity; it suffices to apPly the strong condition to a single pair $(x, y)$ . In Section 1 we show that is abelian or finite-dimensional. We also the two properties are equivalent if $\mathcal{A}$

116

O. BRATTELI, G. A. ELLIOTT and D. W. ROBINSON

show that strong topological transitivity follows from other ergodicity properties, but we do not know if strong topological transitivity is strictly stronger than topological transitivity. In Section 2 we analyze the structure of the action of a compact group $G$ under the assumption that on a strongly transitive $c*$ -system and commute. More specifically, we show that consists of those automorphisms which commute with and which reduce to the identity on of the fixed point algebra of . Then in Section 3 we examine the infinitesimal . In particular we show that if is a closed symmetric deristructure of generates a one-parameter then into vation from the G-finite elements if, and only if, subgroup of commutes with and is zero on the fixed points of . Finally in Section 4 we make some remarks about the generation problem for dissipations. This analysis extends results recently obtained by Kishimoto and Robinson [9], Longo and Peligrad [10], and Robinson, $Strmer$ and Takesaki [11]; see also [1], [2], [3], [8] and [12] for earlier results of a similar nature. $\alpha$


$\alpha$

$\tau$

$\{\alpha_{g} ; g\in G\}$

$\mathcal{A}$

$\beta$

$\tau$

$\mathcal{A}^{\alpha}$

$\alpha$

$(G, \alpha)$

$\delta$

$\delta$

$\mathcal{A}$

$\mathcal{A}_{F}$

$\delta$

$\delta$

$\alpha$

$\tau$

$\alpha$

1. Topological transitivity. In this section we analyze some basic properties of transitivity and strong transitivity as defined in the introduction. First we show that these properties are invariant under the adjunction of an identity. -dynamical system. If does not contain an identity be a Let one can adjoin such an element 1 by a canonical procedure and then extend . to $A=\mathcal{A}+C1$ by setting $C^{*}$


$\mathcal{A}$

$(H, \tau)$

$\tilde{\tau}_{h}(x+\lambda 1)=\tau_{h}(x)+\lambda 1$

-dynamical system without identity and LEMMA 1.1. Let be a the system obtained by adjoining an identity. The following Pairs of conditions are equivalent: is (strongly) topologically transitive; 1. (1s.) $(2s.)$ is (strongly) topologically transrtive. 2. . Identify with its universal representation. This gives a PROOF. faithful representation of J4, and the tensor product Hilbert space gives a rep. Assume resentation of $C^{*}$


$(\tilde{\mathcal{A}}, H,\tilde{\tau})$


$(\hat{\mathcal{A}}, H,\tilde{\tau})$

$ls\Rightarrow 2s$

$\mathcal{A}$

$\tilde{\mathcal{A}}\otimes\tilde{\mathcal{A}}$

$\sum_{i=1}^{n}\tilde{x}_{i}\otimes\overline{y}_{i}\neq 0$

for some

$\tilde{x}_{i},\tilde{y}_{i}\in\tilde{\mathcal{A}}$

. Let

converges weakly to

$\tilde{x}\otimes\tilde{y}$

be an approximate identity of for all . Thus for some

$(e_{\alpha})$

$\tilde{x},\tilde{y}\in_{c}X$

$\sum_{i\Rightarrow 1}^{n}e_{\alpha}\tilde{x}_{i}\otimes\tilde{y}_{i}e_{\alpha}\neq 0$

in


. Hence

there exists

$h\in H$

such that

$\mathcal{A}$

$\alpha$

. Then

$e_{a}\tilde{x}\otimes\tilde{y}e_{a}$

sufficiently large,

$C^{*}$

117

-dynamical systems

$e_{\alpha}( \sum_{i=1}^{n}\tilde{x}_{i}\tau_{h}(\tilde{y}_{i}))\tau_{h}(e_{\alpha})=\sum_{i=1}^{n}e_{a}\tilde{x}_{i}\tau_{h}(\tilde{y}_{i}e_{\alpha})\neq 0$

by Condition ls.

But this implies $\sum_{i=1}^{n}\tilde{x}_{i}\tau_{h}(\tilde{y}_{i})\neq 0$

is fulfilled. . This is evident from the embedding of (JZ, ). Ie2. This follows from the above with $n=1$ . in Lemma 1.1 is useful because it means that one can usually assume that has an identity in the discussion of transitivity. Next note that the use of two elements in the definition of topological transitivity is not particularly significant. In fact by iteration one readily sees that , is topologically transitive if, and only if, for each sequence $h_{k}\in H$ , such that of non-zero elements of there exist

so Condition

$2s$


$2s\Rightarrow ls$

$H,\tilde{\tau}$

$\mathcal{A}$


$x_{1},$

$h_{1},$

$\mathcal{A}$

$h_{2},$

$x_{2},$

$\cdots$

$x_{k}$

$\cdots$

$\tau_{h_{1}}(x_{1})\tau_{\hslash_{2}}(x_{2})\cdots\tau_{h_{k}}(x_{k})\neq 0$

.

A similar conclusion is true for strong topological transitivity.

PROPOSITION 1.2. The following conditions are equivalent: 1. The -dynamical system is strongly topologically transitive. 2. For each family of finite sequences of elements of satisfying $C^{*}$


$\{x\ell^{1)}, \cdots , x\ell^{k)} ; i=1, \cdots , n\}$

$\mathcal{A}$

$\sum_{i=1}^{n}x_{i}^{(1)}\otimes x_{i}^{(2)}\otimes\cdots\otimes x_{i}^{(k)}\neq 0$

in the k-fold algebraic tensor product such that

$\otimes^{k}\mathcal{A}$

, there exist

$h_{1},$

$h_{2},$

$\cdots$

,

$h_{k}\in H$

$\sum_{i=1}^{n}\tau_{h_{1}}(x_{i}^{(1)})\tau_{h_{2}}(x_{i}^{(2)})\cdots\tau_{h_{k}}(x_{i}^{(k)})\neq 0$

PROOF.

.

$2\leqq k0$

$x_{i}$

$3\Rightarrow 2$

$\mathcal{A}$

$\mathcal{A}=M_{2}$

$2\cross 2$

$\tau$

$p,$

$q$

$\sup_{h}\Vert P^{\tau_{\hslash}(q)\Vert0$

.

Thus the

$x_{i}$

are

linearly dependent, which is

a contradiction.

-dynamical system with $H$ compact. COROLLARY 1.6. Let be a The following conditions are equivalent: 1. is ergodic; 2. is topologically transztive; 3. is strongly top0l0gcally transztive. . Since is ergodic there is a unique PROOF. is evident. by given -invariant state over $C^{*}$


$\tau$

$\tau$

$\tau$

$1\Rightarrow 3$

$3\Rightarrow 2\Rightarrow 1$

$\mathcal{A}$

$\omega$

$\tau$

$\tau$

$\omega(x)1=\int_{H}dh\tau_{h}(x)$

But is a trace by [7] and hence tion 3 follows from Theorem 1.5. $\omega$

$\Omega_{\omega}$

.

is separating for

$\pi_{\omega}(\mathcal{A})’’$

.

Thus Condi-

-dynamical system. Assume there exists THEOREM 1.7. Let be a is a -ergodic state such that the corresponding representation is asymptotically abelian in the sense faithful. Moreover assume that $C^{*}$


$(\mathcal{H}_{\omega}, \pi_{\omega}, \Omega_{\omega})$

$\omega$

$\tau$


$inf\Sigma^{n}\Vert\pi_{\omega}([\tau_{\hslash}(x_{i}), y_{i}])\psi_{i}\Vert=0$

$h\in Ht=1$

for

all

finite sequences of elements


and vectors is strongly topologzcally transitive and furthermore $x_{i},$

$y_{i}\in \mathcal{A}$

$\psi_{i}\in \mathcal{H}_{\omega}$

.

It follows that

$C^{*}$

121

-dynamical systems

$\sup_{h\in H}\Vert x\tau_{\hslash}(y)\Vert=\Vert x\Vert\cdot\Vert y\Vert$

,

$x,$

$y\in \mathcal{A}$

.

PROOF. Again assume $\sum_{i=1}^{n}x_{i}\otimes y_{i}\neq 0$

with the

linearly independent and

$x_{i}$

$y_{1}\neq 0$

, but

$\sum_{i=1}^{n}x_{i}\tau_{h}(y_{i})=0$

.

for all

$h\in H$

Then $\sum_{i\Rightarrow 1}^{n}\pi_{\omega}(\tau_{h}(z)x_{i}\tau_{h}(y_{i}y_{1}^{*}z^{*}))=0$

for all

and . Therefore by taking a limit over a suitable net of combinations over one concludes that

$h\in H$

convex

$z\in \mathcal{A}$

$h$

$\sum_{i=1}^{n}\pi_{\omega}(x_{i})\omega(zy_{i}y_{1}^{*}z^{*})=0$

This again follows from the general theory of -invariant states as described in Section 4.3 of [5] together with our choice of the asymptotic abelianness condi. Hence is faithful, can be chosen such that tion. Now since the must be linearly dependent, which is a contradiction. Consequently is strongly topologically transitive. The last statement of the theorem follows by an argument given in [9]. One has $\tau$

$\omega(zy_{1}y_{1}^{*}z^{*})>0$

$z$

$\pi_{\omega}$

$x_{i}$


$\omega((a\tau_{h}(b))^{*}(x\tau_{h}(y))^{*}(x\tau_{h}(y))(a\tau_{h}(b))\leqq\sup_{h}\Vert x\tau_{h}(y)\Vert^{2}\omega((a\tau_{h}(b))^{*}(a\tau_{h}(b)))$

.

It then follows from the conditions of asymptotic abelianness and ergodicity of that

$\omega$

$\omega(a^{*}x^{*}xa)\omega(b^{*}y^{*}yb)\leqq\sup_{\hslash}\Vert x\tau_{h}(y)\Vert^{2}\omega(a^{*}a)\omega(b^{*}b)$

Since

$\pi_{\omega}$

.

is faithful it follows that $\Vert x\Vert^{2}\Vert y\Vert^{2}\leqq\sup_{h}$

I

$x\tau_{h}(y)\Vert^{2}\leqq\Vert x\Vert^{2}\Vert y\Vert^{2}$

which gives the desired conclusion. The property of strong topological transitivity can also be expressed in terms norms on tensor products, . , the system of is strongly topologically transitive if, and only if, $e$


$g.$

$\Vert\sum_{i=1}^{n}x_{i}^{(1)}\otimes x_{i}^{(2)}\otimes\cdots\otimes x_{i}^{(k)}\Vert_{k}=\sup_{\hslash_{1\prime}\cdots.h_{k}\in H}\Vert\sum_{l=1}^{n}\tau_{h_{1}}(x_{i}^{(1)})\tau_{\hslash_{2}}(x_{i}^{(2)})\cdots\tau_{h_{k}}(x_{i}^{(k)})\Vert$

dePnes a norm on

.

This rephrasing follows directly from the original definition for and from Proposition 1.2 for higher . Unfortunately -norms, although this is the case if necessarily not are is norms these $\otimes^{k}\mathcal{A}$

for all

$k\geqq 2$

$k=2$

$k$

$C^{*}$

$\mathcal{A}$

122

$0$

.

BRATTELI, G. A. ELLIOTT and D. W. ROBINSON

abelian by Theorem 1.3.

norm

For example all

$c*$

-norms on


satisfy the

cross-

property $\Vert x\otimes y\Vert=\Vert x\Vert\cdot\Vert y\Vert$

But the norm

.

has this property if, and only if,

$\Vert\cdot\Vert_{2}$

$(*)$

$\sup_{\hslash\in}\Vert x\tau_{h}(y)\Vert=\Vert x\Vert\cdot\Vert y\Vert$

.

fails (see the We have, however, already given an example where property is a necessary condition for the remark after Theorem 1.4). Thus to guarantee to general the more -norms. It is also sufficient be cross-norm $(*)$

$(*)$

$\Vert\cdot\Vert_{k}$

$C^{*}$

property $\Vert a\otimes b\Vert_{k+l}=\Vert a\Vert_{k}\Vert b\Vert_{l}$

. This follows by the argument used in the proof of Propand osition 2.1 of [9]. It would be of interest to obtain necessary and sufficient for the to be $c*$ -norms. It would also be of interest conditions on and the similar norms dePned for norm asymptotically abelian to compare the systems in [9]. These latter norms are defined as above except the supremum is replaced by a limit supremum, and Proposition 2.1 of [9] establishes conditions -norm property is valid. under which the for

$b\in \mathcal{A}_{l}$

$a\in \mathcal{A}_{k}$




$C^{*}$

2. Topological transitivity and compact actions. Next we consider a strongly topologically transitive -system and also an action of a compact group on . We assume commutes with and our aim is to analyze the structure of . $C^{*}$

$G$

$\alpha$

$\mathcal{A}$


$\alpha$

$\tau$

$(G, \alpha)$

-dynamTHEOREM 2.1. Let be a strongly topologically transitive (faithful) continuous action of a compact grouP ical system and as -automorphisms of . If is a -automorphism of such that such that $[\beta, \tau]=0$ $\beta(x)=x$ , the fixed point algebra of , then and for all $C^{*}$


$G$

$\alpha a$

$[\alpha, \tau]=0$

$\mathcal{A}$

$\beta$

$x\in \mathcal{A}^{a}$

for some

$g\in G$

$*$

$\mathcal{A}$

$*$

$\beta=\alpha_{g}$

$\alpha$

.

REMARK. This theorem is a direct generalization of Theorem 1.1 of [11]. If is a von Neumann algebra, the theorem remains true if strong topological transitivity is replaced by ergodicity ([12]; see also [1], [2], [3]). If is abelian, one may also replace strong topological transitivity by topological transitivity ([10], Theorem 3.1). PROOF. Let denote the set of G-finite elements in . . the linear span of the spectral subspaces $\mathcal{A}$

$G$

$\mathcal{A}_{F}$

$\mathcal{A},$

$\mathcal{A}^{a}(U)=\{\int dgTr(U_{g}^{-1})\alpha_{g}(x);x\in \mathcal{A}\}$

$i$

$e$

$c*$

123

-dynamical systems

corresponding to the irreducible representations $U$ of $G$ . Alternatively, is such that the linear span of $\{\alpha_{g}(x);g\in G\}$ is characterized as the set of is a dense -subalgebra of and if finite-dimensional. We note that . is a finite-dimensional -invariant subspace then Next for each $h\in H$ let denote the linear map from into defined by $\mathcal{A}_{F}$

$x\in \mathcal{A}$

$V\subseteqq \mathcal{A}$

$\mathcal{A}$

$*$

$\mathcal{A}_{F}$

$V\subseteqq \mathcal{A}_{F}$

$\alpha$

$T_{h}$

$\tau_{h}(\sum_{i=1}^{n}x_{i}\otimes y_{i})=\sum_{i=1}^{n}x_{i}\tau_{\hslash}(y_{i})$

It follows immediately from

$\mathcal{A}$


$[a, \tau]=0$

.

that

$T_{h}(\alpha_{g}\otimes\alpha_{g})=\alpha_{g}T_{h}$

for all

$g\in G$

, and similarly since

$[\beta, \tau]=0$

,

$T_{h}(\beta\otimes\beta)=\beta T_{h}$

.

be a finite-dimensional -invariant subspace of and introduce the Now let finite-dimensional subspace $W=V+\beta(V)$ . It follows from strong topological transitivity that there is a finite subset of $H$ such that the map $V$

$\mathcal{A}$

$\alpha$

$H_{W}$

$\bigoplus_{\hslash\in H_{W}}T_{h}$

:

$W \otimes Warrow\bigoplus_{h\in H_{W}}\mathcal{A}$

is injective.

OBSERVATION 1.

If

and

$y\in \mathcal{A}_{F}$

$x,$

$\phi,$

$\psi\in \mathcal{A}^{*}$

then

$\int_{G}dg\overline{\psi(\beta a_{g}(x))}\phi(\beta\alpha_{g}(y))=\int_{G}dg\overline{\psi(\alpha_{g}(x))}\phi(\alpha_{g}(y))$

PROOF. After replacing

$x$

by

$x^{*}$

and

by

$\psi$

$\psi*it$

suffices to show

$\int_{G}dg\psi(\beta\alpha_{g}(x))\phi(\beta\alpha_{g}(y))=\int_{G}dg\psi(\alpha_{g}(x))\phi(\alpha_{g}(y))$

Since

$x,$


$\{a_{g}(y);g\in G\}$

.

.

generated by $\{\alpha_{g}(x);g\in G\}$ and -invariant space is finite-dimensional. Set $W=V+\beta(V)$ . The injection

, the linear

$V$

$a$

$\bigoplus_{h\in H_{W}}T_{h}$

transports the linear functional subspace

:

$W \otimes Warrow\bigoplus_{h\in H_{W}}\mathcal{A}$

$\psi\otimes\phi$

on

$W\otimes W$

onto a linear functional

$( \bigoplus_{\hslash\in H_{W}}T_{\hslash})(W\otimes W)\subseteqq\bigoplus_{h\in H_{W}}\mathcal{A}$

$\xi$

on the

.

It then follows from the Hahn-Banach theorem that has a continuous extension which we also denote by . But has a linear decomposition to $\xi$

$\xi$

$\xi$

$\bigoplus_{h\in H_{W}}\mathcal{A}$

$\xi=\bigoplus_{h\in H_{W}}\xi_{h}$

Therefore

.

124

$0$

.


$\int_{G}dg\psi(\beta a_{g}(x))\phi(\beta a_{g}(y))=\int_{G}dg(\psi\otimes\phi)(\beta\alpha_{g}(x)\otimes\beta\alpha_{g}(y))$

$= \int_{G}dg\sum_{\hslash\in H_{W}}\xi_{h}(T_{h}(\beta a_{g}(x)\otimes\beta\alpha_{g}(y)))$

$= \int_{G}dg\sum_{h\in H_{W}}\xi_{h}(\beta a_{g}(x\tau_{h}(y)))$

$= \sum_{h\in H_{W}}\xi_{h}(\beta(\int_{G}dg\alpha_{g}(x\tau_{h}(y)))$

,

$= \sum_{\hslash\in H_{W}}\xi_{\hslash}(\int_{G}dga_{g}(x\tau_{\hslash}(y)))$

$= \int_{G}dg\psi(a_{g}(x))\phi(a_{g}(y))$

leaves the fixed points of where the penultimate step uses the fact that pointwise invariant, and the ultimate step follows from reversal of the previous $\mathcal{A}^{\alpha}$

$\beta$

$a$

steps.

denote the G-finite elements for the action of as Next let right (or, equivalently, left) translations on $C(G)$ . Thus $C_{F}(G)$ is the set of whose orbit under right translations spans a finitecontinuous functions over dimensional subspace of $C(G)$ . Again $C_{F}(G)$ is a dense -subalgebra of $C(G)$ . $G$

$C_{F}(G)\subseteqq C(G)$

$G$

$*$

OBSERVATION 2. Every

has the

$f\in C_{F}(G)$

form

$f(g)= \sum_{l=1}^{n}\phi_{i}(\alpha_{g}(x_{i}))$

where PROOF.

. and be the subspace of Let

$x_{i}\in \mathcal{A}_{F}$

$\phi_{i}\in \mathcal{A}^{*}$

$\mathcal{D}$

$C_{F}(G)$

of functions of the form

$f(g)= \sum_{i=1}^{n}\phi_{i}(a_{g}(x_{i}))$

In the proof of Observation 1 we established an identity of the form $\psi(a_{g}(x))\phi(\alpha_{g}(y))=\sum_{h\in H_{W}}\xi_{h}(a_{g}(x\tau_{h}(y)))$

and is a finite subset of $H$ and . The , and depending on the finite-dimensional subspace spanned by the orbits , of , and . Since and commutes with it follows that . Therefore this identity establishes that is an algebra. Also

for all

$x,$


$\psi,$

$H_{W}$

$\xi_{\hslash}\in \mathcal{A}^{*}$

$\phi\in \mathcal{A}^{*}$

$V$

$\alpha_{g}(y)$

$x$

$x,$

$y$


$\alpha_{g}(x)$

$\alpha$

$\tau$

$x\tau_{h}(y)\in \mathcal{A}_{F}$

$\mathcal{D}$

$\overline{\psi(\alpha_{g}(x))}=\psi^{*}(\alpha_{g}(x^{*}))$

so

is a -algebra. As

is a faithful representation of

it follows that the separate points of , and hence is dense in $C(G)$ by the Stonefunctions in is closed under regularization by matrix elements Weierstrass theorem. Since $\mathcal{D}$

$*$

$a$

$G$

$\mathcal{D}$

$\mathcal{A}_{F}$

$\mathcal{D}$

$G$

$C^{*}$

-dynamical systems

125

of the irreducible representations of it follOws that has the same property (with respect to the right regular representation). Then it easily follows from the orthogonality relations that $\mathcal{D}=C_{F}(G)=the$ linear span of the matrix elements of the irreducible representations of . $G$

$\mathcal{D}$

$G$

OBSERVATION 3. There exis an isometric linear isomorphism with the Properties $ts$

1.

$B( \sum_{i=1}^{n}\phi_{i}(\alpha_{g}(x_{i})))=\sum_{i=1}^{n}\phi_{i}(\beta a_{g}(x_{i}))$

2. 3. 4.

$B(f_{1}f_{2})=B(f_{1})B(f_{2})$ $B(\overline{f})=\overline{B(f)}$

,

,

$f_{1},$

,

$B(r_{g}f)=r_{g}(B(f))$

$x_{i}\in \mathcal{A}_{F},$

,

$\phi_{i}\in \mathcal{A}^{*}$

$f_{2}\in C_{F}(G)$

$f\in C_{F}(G)$

,

$f\in C_{F}(G)$

,

$B;C_{F}(G)arrow C_{F}(G\rangle$

,

,

where denotes right translations. PROOF. It follows from Observations 1 and 2 that there exists a unitary operator on $L^{2}(G)$ with the action given by Property 1. Thus is well defined as a linear operator from $C_{F}(G)$ into $C(G)$ by Observation 2. But since , Observation 2 implies that $B$ is in fact an operator from $C_{F}(G)$ into $C_{F}(G)$ . Now if $m(f)$ denotes the operator of multiplication by $f\in C_{F}(G)$ on $L^{2}(G)$ then $r$

$B$

$B$

$\phi_{i}(\beta\alpha_{g}(x_{i}))=(\beta^{*}\phi_{i})(\alpha_{g}(x_{i}))$

$(B^{*}F, m(f)B^{*}G)=(F, m(Bf)G)$

for all

$F,$

$G\in L^{2}(G)$

. Since

$B$

is unitary on

$L^{2}(G)$

it then follows that

$\Vert f\Vert_{\infty}=\Vert m(f)\Vert=\Vert m(Bf)\Vert=\Vert Bf\Vert_{\infty}$

.

,

is an isometry from $C_{F}(G)$ into $C_{F}(G)$ . By considering instead of we see that maps $C_{F}(G)$ onto $C_{F}(G)$ . The multiplicative property of follows from the calculation $i.e$

$B$

$\beta^{-1}$

$B$

$B$

$B( \psi(\alpha_{g}(x))\phi(a_{g}(y)))=B(\sum_{h\in H_{W}}\xi_{h}(\alpha_{g}(x\tau_{h}(y))))$

$= \sum_{h\in H_{W}}\xi_{\hslash}(\beta a_{g}(x\tau_{h}(y)))$

$=\psi(\beta a_{g}(x))\phi(\beta a_{g}(y))$

$=B(\psi(a_{g}(x)))B(\phi(a_{g}(y)))$

and

$B$

commutes with the involution because $B(\overline{\psi(\alpha_{g}(x))})=B(\psi^{*}(\alpha_{g}(x^{*})))$

$=\psi^{*}(\beta a_{g}(x^{*}))$

$=\overline{\psi(\beta\alpha_{g}(x))}=\overline{B\psi(a_{g}(x))}$

Finally

$B$

commutes with right translations because

.

$\beta$

O. BRATTELI, G. A. ELLIOTT and D. W. ROBINSON $B(r_{h}(\psi(\alpha_{g}(x))))=B(\psi(\alpha_{g}(a_{h}(x))))$

$=\psi(\beta\alpha_{g}(a_{h}(x)))$

$=r_{h}(\psi(\beta a_{g}(x)))=r_{h}(B(\psi(\alpha_{g}(x))))$

.

The proof of Theorem 2.1 is now straightforward. The operator as defined is an isometry from $C_{F}(G)$ onto $C_{F}(G)$ . But as $C_{F}(G)$ by continuity to an isometry is norm dense in $C(G)$ one can extend from $C(G)$ onto $C(G)$ . The properties established in Observation 3 then extend by continuity. Hence is a -automorphism of $C(G)$ which commutes with right of $C(G)$ cortranslations. Now let be the homeomorphism of the spectrum $b(e)=g$ responding to . If then $B$

$B$

$B$

$*$

$G$

$b$

$B$

$b(h)=b(eh)=b(e)h=gh$

for all

$h\in G,$

$i.e$

.

$B$

is left translation by

$g^{-1}$

.

Thus

$\psi(\beta\alpha_{h}(x))=B(\psi(\alpha_{h}(x)))=\psi(\alpha_{g}\alpha_{h}(x))$

$for_{L}^{-}al1x\in \mathcal{A}_{F},$


, and

$h\in G$

.

Consequently $\beta(x)=a_{g}(x)$

. which implies for all The above procedure of constructing mation about the spectral subspaces of $\beta=a_{g}$

$x\in \mathcal{A}_{F}$

$C_{F}(G)$

from elements of

$(\mathcal{A}, G, \alpha)$

$\mathcal{A}_{F}$

gives infor-

.

be a strongly topologically transitive -dynamCOROLLARY 2.2. Let (faithful) as -autocontinuous action of a compact group a ical system and . Further let denote the set of irredusuch that morphisms of by cible representations of and for each define the spectral subspace $C^{*}$


$G$

$a$

$[\alpha, \tau]=0$

$\mathcal{A}$

$G$

$\mathcal{U}(G)$

$U\in \mathcal{U}(G)$

$\mathcal{A}^{a}(U)$

$\mathcal{A}^{a}(U)=\{\int dgTr(U_{g}^{-1})\alpha_{g}(x);x\in \mathcal{A}\}$

It

$\vee follows$

1. 2. If

$*$

.

that

$\mathcal{A}^{\alpha}(U)\neq\{0\},$

$U_{1},$

$U\in \mathcal{U}(G)$

$U_{2}\in \mathcal{U}(G)$

and

,

$V\in \mathcal{U}(G)$

occurs in the decomposition of

$(\mathcal{A}^{a}(U_{1})\mathcal{A}^{a}(U_{2}))\cap \mathcal{A}^{a}(V)\neq\{0\}$

$U_{1}\otimes U_{2}$

then

.

We conclude this section with two examples which demonstrate the difficulty . In both examples one has a -autoin characterizing the automorphisms morphism which leaves invariant each finite-dimensional -invariant subspace $*$

$\alpha_{g}$

$\alpha$

$\beta$

of

$\mathcal{A}$

but

$\beta\not\in\alpha_{G}$

.

be the permutation group EXAMPLE 2.3 (Longo and Peligrad [10]). Let with the has order 6 and is generated by two elements on 3 elements; $S_{3}$

$S_{3}$

$r,$

$s$

$C^{*}$

relations $r^{3}=e,$ with dim

$s^{2}=e,$

$rs=sr^{2}$

$\gamma_{1}=\dim\gamma_{2}=1$

$\gamma_{8}$

127

-dynamical systems

. The dual

, and dim

$\gamma_{3}=2$

consists of 3 representations , where $\hat{S}_{3}$

$\gamma_{1}(r)=1$

,

$\gamma_{1}(s)=1$

$\gamma_{2}(r)=1$

,

$\gamma_{2}(s)=-1$

$\gamma_{3}(r)=(\begin{array}{ll}\cos 2\pi/3 \sin 2\pi/3-\sin 2\pi/3 \cos 2\pi/3\end{array})$

,

$\gamma_{3}(s)=(\begin{array}{l}-100 1\end{array})$

$\gamma_{1},$

$\gamma_{2}$

,

.

be the algebra of Let . Then matrices and define is ergodic, and hence strongly topologically transitive by Theorem 1.4. The representation of has the decomposition $2\cross 2$

$\mathcal{A}=M_{2}$

$\alpha$

$\alpha_{g}=Ad(\gamma_{3}(g))$

$\alpha$

$G$

$\alpha=\gamma_{1}\oplus\gamma_{2}\oplus\gamma_{3}$

. such that into irreducibles and there is a unitary operator up requirement that to a phase factor by the is determined The operator . Moreover is in the -subspace of . Let $\beta=Ad(V)$ . It follows that occurs with multiplicity one it follows as each of the representations are the subspaces corresponding to that the only -invariant subspaces of representations, all and linear combinations of these subspaces. these three Therefore leaves all these a-invariant subspaces invariant, but nevertheless $\alpha_{g}(V)=\gamma_{2}(g)V$

$V\in \mathcal{A}$

$V$

$V$

$[\beta, \alpha]=0$

$\mathcal{A}$

$\gamma_{2}$

$\gamma_{1},$

$\gamma_{2},$

$\gamma_{3}$

$\mathcal{A}$

$\alpha$

$\beta$


.

denote the continuous functions over the two EXAMPLE 2.4. Let sphere and the canonical action of the group $G=SO(3)$ of rotations on . is topologically transitive, and hence strongly topologically The system corresponding transitive by Theorem 1.3. Let be the -automorphism of . Nevertheless $[\beta, a]=0$ and a calto reflection about the origin. Then culation with spherical harmonics shows that all the irreducible representations with multiplicity one and hence of $SO(3)$ occur in the decomposition of leaves all the finite-dimensional -invariant subspaces invariant. $\mathcal{A}=C(S_{2})$

$S_{2}$

$\mathcal{A}$

$\alpha$

$(\mathcal{A}, G, a)$

$C(S_{2})$

$*$

$\beta$


$\alpha$

$\beta$

$\alpha$

3. The infinitesimal structure of

$(G, a)$

.

The next result gives an infinitesimal characterization of the one-parameter subgroups of the group action considered in Theorem 2.1. It is similar to Theorem 2.1 of [11]. It is possible to give a proof roughly following the lines of the proof of Theorem 2.1 of [11], but we give a shorter proof based upon the relation between and $C_{F}(G)$ established in the previous section. $(G, \alpha)$

$\mathcal{A}_{F}$

be a strongly topologically transitive -dynamTHEOREM 3.1. Let as -autoical system and a (faithful) continuous action of a compact group . Further let be a symmetric derivation of such that morphisms of $C^{*}$


$G$

$a$

$\mathcal{A}$

$[\alpha, \tau]=0$

$\delta$

$*$

128

$0$

.


, the G-finite elements of . The following conditions with domain are equivalent: . is closable and its closure generates one-parameter subgroup of 1. $\delta(x)=0$ , , 2. . $D(\delta)=\mathcal{A}_{F}$

$\mathcal{A}$

$\mathcal{A}$

$\delta$

$\delta$

$b$

$a$

$\alpha_{G}$

$x\in \mathcal{A}^{\alpha}$

$a$

.

$\delta\tau_{h}(x)=\tau_{h}\delta(x)$

,

$x\in \mathcal{A}_{F},$

.

$h\in H$

REMARKS. 1. The drawback of this result as opposed to the comparable results of [9] and [11] is that in Condition 2 we must explicitly assume that . In [9] and [11] this was a consequence is zero on the fixed point algebra , simplicity of , and asymptotic abelianness of . Also this of is abelian. holds if If is abelian, the techniques of [10] show that the assumption of strong 2. topological transitivity may be replaced by topological transitivity. then using the and is evident. . If PROOF. is the a-invariant span of notation of Section 2 with $W=V+\delta(V)$ where and , one calculates that $\delta$


$D(\delta)=\mathcal{A}_{F}$


$\mathcal{A}$

$\mathcal{A}$

$G$

$2\Rightarrow 1$

$1\Rightarrow 2$

$x,$


$\phi,$


$V$

$x$

$y$

$\int dg\{\psi(\alpha_{g}(x))\phi(\delta(\alpha_{g}(y)))+\psi(\delta(\alpha_{g}(x)))\phi(\alpha_{g}(y))\}$

$= \int dg\sum_{h\in H_{W}}\xi_{h}(\alpha_{g}(x)\tau_{h}(\delta\alpha_{g}(y))+\delta(\alpha_{g}(x))\tau_{h}(\alpha_{g}(y)))$

$= \sum_{\hslash\in H_{W}}\xi_{\hslash}(\int dg\delta(a_{g}(x\tau_{\hslash}(y))))$

$=0$

.

The last step follows because is zero on define a linear operator $D:C_{F}(G)arrow C(G)$ by $\delta$


.

This establishes that one can

$D( \sum_{i=1}^{n}\phi_{i}(a_{g}(x_{i})))=i\sum_{i=1}^{n}\phi_{i}(\delta(a_{g}(x_{i})))$

and $D$ is symmetric on $L^{2}(G)$ . (Use Observation 2 of Section 3.) But further calculations analogous to those in the proof of Observation 3 of Section 2 then establish that $D(f_{1}f_{2})=D(f_{1})f_{2}+f_{1}D(f_{2})$

$D(\overline{f})=\overline{D(f)}$

,

$D(r_{g}f)=r_{g}(D(f))$

,

,

$f_{1},$

$f_{2}\in C_{F}(G)$

$f\in C_{F}(G)$

,

$f\in C_{F}(G)$

.

,

Since $D$ commutes with right translations it leaves the corresponding finitedimensional spectral subspaces of $C(G)$ invariant. Hence $D$ is essentially selfadjoint since it is the direct sum of bounded symmetric operators. Consequently the closure of $D$ generates a strongly continuous one-parameter group of -automorphisms of $C(G)$ (see, for example, the discussion in Example 3.2.67 of $\overline{D}$

$\beta$

$*$

C’-dynamical systems

129

must commute with right translations. Therefore is a one-parameter subgroup of left translations by the argument used in the proof [5]).

Moreover

$\beta$

$\beta$

of Theorem 2.1. Let denote left translations on $t-h_{t}$ of such that $l$

$C(G)$

; then there is a one-parameter subgroup

$G$

$\beta_{t}\phi(\alpha_{g}(x))=l_{h_{t}}\phi(\alpha_{g}(x))=\phi(\alpha_{h_{t}}^{-1}\alpha_{g}(x))$

for . But if and group of -automorphisms $x\in \mathcal{A}_{F}$

$\hat{\delta}$


$*$

$trightarrow a_{h_{t}}^{-1}$

denotes the generator of the one-parameter

one finds

$\phi(\delta(\alpha_{g}(x)))=D\phi(\alpha_{g}(x))=\phi(\hat{\delta}(a_{g}(x)))$

is invariant under Thus on . Finally since is a core for ; see Chapter 3 of [5]. Hence is for all it follows that is only assumed to be . (Compare [6], where the group closable and locally compact, but the derivation $D$ is assumed to be closed.)

by differentiation.

$\delta=\hat{\delta}$

$\mathcal{A}_{F}$

$\mathcal{A}_{F}$

$\alpha_{h_{t}}$

$\hat{\delta}$

$t$

$\delta$

$\mathcal{A}_{F}$

$G$

$\overline{\delta}=\hat{\delta}$

REMARK. The generation property in Theorem 2.1 of [11] was established by first proving that leaves invariant each finite-dimensional a-invariant subspace. This can be deduced directly from strong topological transitivity as follows. , be a basis of linearly independent elements of the Let by a matrix, invariant subspace . Then the action is given on $\delta$

$x_{1},$

$x_{2},$

$\cdots$

$\alpha-$

$x_{n}$

$\mathcal{M}$

$a$

$\mathcal{M}$

$\alpha_{g}(x_{i})=\sum_{j\Rightarrow 1}^{n}U_{ji}(g)x_{j}$

,

and by linear rearrangement, using the orthogonality relations in the group, one is in fact unitary. Then since commutes with one can suppose that calculates that $(U_{ji})$

$a$

$a_{g}( \sum_{i=1}^{n}x\beta\tau_{h}(x_{i}))=\sum_{i=1}^{n}xf\tau_{h}(x_{i})$

.

Thus $0= \delta(\sum_{i=1}^{n}xf_{T_{h}}(x_{i}))=\sum_{i=1}^{n}\delta(x_{i})^{*}\tau_{\hslash}(x_{i})+xf\tau_{h}(\delta(x_{i}))$

for all

. But

$h\in H$

strong topological transitivity then implies that $\sum_{i=1}^{n}\delta(xf)\otimes x_{i}+x\beta\otimes\delta(x_{i})=0$

Consequently $\sum_{i\Leftarrow 1}^{n}\delta(x_{i})\omega_{j}(x\beta)=-\sum_{i=1}^{n}x_{i}\omega_{j}(\delta(xf))\in \mathcal{M}$

for any state

$\omega_{j}$

over

$\mathcal{A}$

or, by linear algebra, $\delta(x_{i})Det\overline{\omega_{j}(x_{i})}\in \mathcal{M}$

.

$\tau$

130


are linearly independent the But since the . determinant is non-zero and hence $x_{i}$

can be chosen such that the

$\omega_{j}$

$\delta(x_{i})\in \mathcal{M}$

4. Dissipations. We conclude with some remarks on the generation problem for dissipations. Consider the assumptions of Theorem 3.1 but with a symmetric dissipation, $i.e$ . $\delta$

$\delta(x^{*}x)\leqq x^{*}\delta(x)+\delta(x)^{*}x$

for all

.

It is then natural to ask whether the conditions and imply that is closable and its closure generates a strongly positive semigroup . If is simple with identity, is is norm asymptotically abelian, and abelian, then this question is answered in the affirmative by Theorem 3.1 of [9] and in fact is completely positive. On the other hand the example in Section , $G=H=Z_{2}\cross Z_{2},$ , and $\delta=H$ shows that even if 3 of [4] with generates a positive semigroup strong topological transitivity of does not necessarily imply that the semigroup is completely positive. If one tries to tackle this problem with the techniques of the present paper then it is not clear that the dissipation lifts to an operator $D$ on $C_{F}(G)$ , as in the proof of Theorem 3.1, $i.e$ . by the definition $x\in \mathcal{A}_{F}$

$\delta(\mathcal{A}^{\alpha})=\{0\}$

$[\delta, \tau]=0$

$\delta$

$\overline{\delta}$

$\beta$

$G$

$\mathcal{A}$

$\tau$

$\beta$

$\mathcal{A}=M_{2}$

$\delta$

$\tau=\alpha$


$\delta$

$D(\phi(a_{g}(x)))=\phi(\delta(a_{g}(x)))$

,



.

is abelian and is only assumed to be topologically transitive then it follows from the techniques used in the proof of Theorem 3.1 in [10], such that combined with techniques of [3], that there exists a function If, however,

$G$

$\tau$

$\phi:\hat{G}rightarrow C$

$\delta(x)=\phi(\gamma)x$

for all $\gamma\in\hat{G}$

.

, the a-spectral subspace of corresponding to the character Using this, it is easy to see that $D$ is well defined, and in fact is given by $\mathcal{A}$

$x\in \mathcal{A}^{\alpha}(\gamma)$

$D(\gamma(g))=\phi(\gamma)\gamma(g)$

.

But $D$ is not generally a dissipation, or, equivalently, is not generally negative dePnite. This is clear from the example on mentioned $D$ is a dissipation if is a complete dissipation, and above. In this example then is generally a generator of a completely positive semigroup (see, for

for

$\gamma\in\hat{G}$

$\phi$

$\mathcal{A}=M_{2}$

$\delta$

$\delta$

example, [3], [4]).

There is one important special case where are abelian. and both $\mathcal{A}$

$G$

$D$

is a dissipation, the case that

$C^{*}$

131

-dynamical systems

-dynamical 1 op0l0gically transitive PROPOSITION 4.1. Let be system where is abelian, and let be a faithful continuous action of a compact abelian group as -automorphisms of such that . Further, let be , the G-finite elements of . a symmetnc operat0r on with domain Assume that . is a $dis\alpha pation,$ $i.e$ . $(\mathcal{A}, H, \tau)$

$\mathcal{A}$

$C^{*}$

$a$

$a$

$G$

$*$

$[\alpha, \tau]=0$

$\mathcal{A}$

$\delta$

$\mathcal{A}$

$D(\delta)=\mathcal{A}_{F}$

$\mathcal{A}$

$\delta$

$i$

$\delta(x^{*}x)\leqq\delta(x)^{*}x+x^{*}\delta(x)$

for ii. iii.

all

$x\in \mathcal{A}_{F}$

for

$\delta(x)=0$

,

all

for

$\delta\tau_{h}(x)=\tau_{h}\delta(x)$

It

$x\in \mathcal{A}^{\alpha}$

,

all


.

$h\in H$

is closable and its closure generates one-parameter semigroup follows of completely posjtive contractions. Furthermore, there exists a convolution semigroup of pr0bability measures on such that that

$\delta$

$\delta$

$a$

$t\geqq 0-\Rightarrow\exp\{-t\delta\}$

$G$

$t\geqq 0-\mu_{t}$

$e^{-t\overline{\delta}}(x)= \int_{G}d\mu_{t}(g)a_{g}(x)$

. all PROOF. The proof combines the tensor product characterization of topologitransitivity on abelian -algebras in Theorem 1.3 with ideas from the proof cal of Theorem 3.1 in [9], but the present case is simpler. has an identity. Next we First note that by Lemma 1.1 we may assume have already remarked that topological transitivity of implies the existence of a function $\phi:Garrow C$ such that

for

$x\in \mathcal{A}$

$C^{*}$

$\mathcal{A}$

$\tau$

$\delta(x)=\phi(\gamma)x$

for all

$x\in \mathcal{A}^{\alpha}(\gamma),$

$x= \sum\tau_{h_{i}}(x_{i})$

$\gamma\in\hat{G}$

.

Next for

$x_{i}\in \mathcal{A}^{\alpha}(\gamma_{i})$

and

$h_{i}\in H,$

$i=1,2,$

$\cdots$

,

; then $\delta(x^{*})x+x^{*}\delta(x)-\delta(x^{*}x)=\sum_{i.j=1}^{k}M_{ij}\tau_{\hslash_{i}}(x_{i})^{*}\tau_{\hslash_{j}}(x_{j})\geqq 0$

where $M_{ij}=\overline{\phi(\gamma_{i})}+\phi(\gamma_{j})-\phi(\gamma_{j}-\gamma_{l})$

Now, if

$y_{i}^{(j)}$

are elements in

$\mathcal{A}$

.

such that

$\tau_{\hslash}(\sum_{i=1}^{n}y\mathfrak{l}^{1)}\otimes yf^{2)}\otimes\cdots\otimes yj^{k)})$

$= \sum_{i=1}^{n}\tau_{\hslash_{1}}(yl^{1)})\tau_{h_{2}}(yj^{2)})\cdots\tau_{\hslash_{k}}(yl^{k)})$

$\geqq 0$

for all

$h\in H^{k}$

,

then it follows from the isometric nature of the morphism $\bigoplus_{h\in H^{k}}T_{h}$

:

$\otimes^{k}\mathcal{A}arrow\bigoplus_{h\in H^{k}}\mathcal{A}$

$k$

, set

132


(Theorem 1.3) that $\sum_{i=1}^{n}yj^{1)}\otimes y\}^{2)}\otimes\cdots\otimes y1^{k)}\geqq 0$

-algebra completion of the algebraic tensor product above, we deduce that Applying this to the inequality for

in

$(\otimes^{k}\mathcal{A})^{\wedge}$

, the

$C^{*}$

$\otimes^{k}\mathcal{A}$

.

$M_{ij}$

$\sum_{i.j=1}^{k}M_{ij}X_{i}^{*}X_{j}\geqq 0$

$(*)$

in position. As

$(\otimes^{k}\mathcal{A})^{\wedge}$

, where

with the

$X_{i}=1\otimes 1\otimes\cdots\otimes x_{i}\otimes\cdots\otimes 1$

is abelian there exist pure states by , replacing taking in product state we find $\mathcal{A}$

$x_{i}\neq 0$

$\omega_{i}$

$(*)$

$\lambda_{i}x_{i}/\omega_{i}(x_{i})$

$x_{i}$

on

$x_{i}$

with , where $\mathcal{A}$

occurring in the

.

Hence, , and applying the

$|\omega_{i}(x_{i})|=\Vert x_{i}\Vert$

$\lambda_{i}\in C$

$i’ th$

$\omega_{1}\otimes\cdots\otimes\omega_{k}$

$\sum_{i.j=1}^{k}M_{ij}\overline{\lambda}_{i}\lambda_{j}\geqq 0$

.

is negative definite on . The rest of the proof is exactly as in the last part of the proof of Theorem 5.1 in [3]. Finally we note that if satisfies the strong condition

Thus the function

$\hat{G}$

$\phi$


$\sup_{h\in H}\Vert x\tau_{h}(y)\Vert=\Vert x\Vert\cdot\Vert y\Vert$

,

$x,$

$y\in \mathcal{A}$

of transitivity, and if $(G, a)$ is the action of a compact abelian group which and commutes with , and if is a symmetric dissipation satisfying , then one can deduce that the associated function satisfies $\delta$

$\delta(\mathcal{A}^{\alpha})=\{0\}$

$\tau$

$\phi:\hat{G}arrow C$

$[\delta, \tau]=0$

$\sqrt{|\phi(\gamma+\mu)|}\leqq\sqrt{|\phi(\gamma)|}+\sqrt{|\phi(\mu)|}$

$(*)$

.

This follows by examining the inequalities $\omega(x^{*}\delta(x)+\delta(x)^{*}x-\delta(x^{*}x))\geqq 0$

, and for states where and elements . is sufficient to ensure that exp It is an interesting question whether to is contractive as a map from since this would ensure that it extends positive semigroup. to a $\lambda\in C,$

$x=\lambda x_{\gamma}+\tau_{h}(x_{\mu})$

$\omega$

$x_{\gamma}\in \mathcal{A}^{\alpha}(\gamma)$

$x_{\mu}\in \mathcal{A}^{\alpha}(\mu)$

$(*)$

$\mathcal{A}_{F}$

$\{-t\delta\}$

$\mathcal{A}_{F}$

References [1]

[2]

H. Araki, R. Haag, D. Kastler and M. Takesaki, Extension of KMS states and chemical potential, Commun. Math. Phys., 53 (1977), 97-134. H. Araki and A. Kishimoto, Symmetry and equilibrium states, Commun. Math. Phys.,

52

(1977),

211-232.

O. Bratteli and D. Evans, Dynamical semigroups commuting with compact abelian actions, Ergodic Theory and Dynamical Systems, 3 (1983), 187-217. [4] O. Bratteli, P. E. T. $J\phi rgensen$ , A. Kishimoto and D. W. Robinson, A -algebraic

[3]

$C^{*}$

$C^{*}$

[5] [6]

133

-dynamical systems

Schoenberg theorem, Ann. Inst. Fourier (to appear). 0. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, Springer-Verlag, New York-Heidelberg-Berlin, 1979. F. Goodman, Translation invariant closed -derivations, Pacific J. Math., 97 (1981), $*$

403-413. [7]

[8]

[9] [10]

R. H egh-Krohn, M. B. Landstad and E. St rmer, Compact ergodic groups of automorphisms, Ann. of Math., 114 (1981), 75-86. A. Kishimoto, On invariant states and the commutant of a group of quasifree automorphisms of the CAR algebra, Rep. Math. Phys., 15 (1979), 21-26; II, Rep. Math. Phys., 15 (1979), 195-198; III, Rep. Math. Phys., 16 (1979), 121-124. A. Kishimoto and D. W. Robinson, Dissipations, derivations, dynamical systems, and asymptotic abelianness, Canberra preprint, 1983. R. Longo and C. Peligrad, Non commutative topological dynamics and compact -algebras, preprint, 1983. actions on -algebras D. W. Robinson, E. St rmer and M. Takesaki, Derivations of simple tangential to compact automorphism groups, Canberra preprint, 1983. M. Takesaki, Fourier analysis of compact automorphism groups (An application of Tannaka duality theorem), in D. Kastler ed. Alg\‘ebres d’op\’erateurs et leurs applications en physique math\’ematique, \’Editions CNRS, Paris, 1979. P. Walters, An Introduction to Ergodic Theory, GTM 79, Springer-Verlag, New York-Heidelberg-Berlin, 1981. $C^{*}$

[11] [12]

[13]

$C^{*}$

Ola BRATTELI

GeOrge A. ELLIOTT

Institute of Mathematics University of Trondheim N-7034 Trondheim-NTH Norway

Mathematics Institute University of Copenhagen Universitetsparken 5 DK-2100, Copenhagen Denmark

Derek W. ROBLNSON Department of Matnematics Institute of Advanced Studies Australian National University Canberra, A. C. T.

Australia

$\emptyset$