数理解析研究所講究録980 巻 1997 年 32-34

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On the Time-Independence of Entropy dimensions
associated with a-Dynamical System$W^{*}$

Shigeo AKASHI
Department of Mathematics, Faculty of Science, Niigata University8050, 2-nomachi, Igarashi, Niigata-shi, 950-21 JAPANand denote the set of all positive integers, the setThroughout this paper,of all real numbers and the set of all complex numbers, respectively, anddenotea separable Hilbert space and the algebra of all bounded operators on , respectively.
Let
be the set of all normal states on. If is a weak* compact
and convex subset of, then the set of all extremal points belonging to ,which is denoted by $exS$ , is non-empty. For any normal state, if there exist both anon-negative sequencesatisfying
and a sequence of normal states, which enable to be represented by the following countable convex
combination:$\mathbb{R}$$\mathrm{N},$$\mathbb{C}$$\mathcal{H}$$B(\mathcal{H})$$\mathcal{H}$$N_{*,+,1}(g(\mathcal{H}))$$S$$B(\mathcal{H})$$S$$\Lambda^{\Gamma_{*,+,1}}(B(\mathcal{H}))$$\phi\in S$$\{\lambda_{k};k\in \mathrm{N}\}$$\{\phi_{k};k\in \mathrm{N}\}\subset exS$$\Sigma_{k}\lambda_{k}=1$$\phi$$\phi=\sum_{k=}^{\infty}1\lambda_{k}\phi_{k}$,then, we define $D(\phi, S)$ by the set of all non-negative sequences that enable to berepresented by the above way. Now, for any positive number, Ohya’s-entropyof is defined by$\phi$$(S, \alpha)$$\alpha\neq 1$$\phi$$S( \emptyset, s, \alpha)=i.nf\{\frac{log\Sigma_{k1k}^{\infty}=\lambda^{\alpha}}{1-\alpha};\{\lambda_{k};k\in N\}\in D(\phi, s)\}$Here, Ohya’s -entropy dimension of$S$$\phi$.

is defined by$d(\phi, S)=inf\{\alpha>0;S(\phi,S, \alpha)<\infty\}$.

Throughout this paper, we will treat the case that $S=N_{*,+,1}(B(\mathcal{H}))$ holds and we will
abbreviate
tofor simplicity.
Letbe a-dynamical system, and be a surjective continuous
action defined on with values in the set of all surjective -automorphism group onthat is, for any
holds and is a surjective *-homomorphism
defined onwith values inwhich is continuous in the -weak operator topology
and satisfies the following condition:$d( \phi,\bigwedge_{*}^{(},+,1(B(\mathcal{H})))$$(B(\mathcal{H}), \mathbb{R}, \alpha)$$d(\phi)$$W^{*}$$\alpha$$*$$\mathbb{R}$$B(\mathcal{H})_{1}$$s,$ $t\in \mathbb{R},$$g(\mathcal{H})$$l_{\dot{i}}m_{\mathrm{t}arrow_{S}}<x,$$\alpha_{s}0\alpha‘=\alpha_{s+}t$$\alpha_{t}$$g(\mathcal{H})$$\alpha_{t}(A)y>=<x,$ $\alpha_{s}(A)y>$$\sigma$,$x,$$y\in \mathcal{H}$,$A\in B(\mathcal{H})$.

Then, it follows from the following theorem that the entropy dimensions of the normalstates constructed by the combination with the initial states and the continuous action

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assocated with the given$W^{*}$-dynamical system are time-independent.

be a surjective continuous action. Then, for any normal state
Theorem. Let.
holds for any$\alpha$$\phi,$$d(\phi)=d(\phi 0\alpha_{t})$$t\in \mathbb{R}$Proof. For any
is defined by$x\in \mathcal{H}$, the vector state constructed by , which is denoted by$x$$\omega_{x}(A)=<x|A|x>$

Here, we can assume that$\phi$,$A\in B(\mathcal{H})$.

is represented by$\rho$$=$$\sum_{k=1}^{\infty}\lambda k|fk><fk|$$\phi(A)$$=$$tr(\rho A)$,$\{\lambda_{k}\}$,$A\in B(\mathcal{H})$is a non-negative sequence satisfyingwheresystem of . Then, can be represented by$\mathcal{H}$$\omega_{x}$$\Sigma_{k}\lambda_{k}--1$,, and$\{f_{k}\}$is an orthonormal$\phi$$\phi=\sum_{k=1}^{\infty}\lambda_{k}\omega \mathrm{e}_{k}$.

holds , $j\neq k$ implies that
implies thatSince
holds. Therefore, it is sufficient to prove that, for any positive integer
holds. Let be an element ofandto) be the cyclic representation of(resp.
be an operator on
constructed by (resp. ). Let
defined byvalues in$\emptyset=0$$\phi 0\alpha_{t}=0$$\omega_{\mathrm{e}_{f}}0\alpha t\neq\omega_{e_{k}}0\alpha_{\mathrm{t}}$$k,$$ex\Lambda’*,+,1(\beta(\mathcal{H}))$$ex\Lambda^{\Gamma_{*,+,1}}(B(\mathcal{H}))$$\omega$$\{\mathcal{H}_{\omega}, \pi_{(v}x\})\omega$$\psi$$\psi$$B(\mathcal{H})$$\{\mathcal{H}_{\psi,\emptyset,\psi}\pi X\}$$\omega$$\omega_{e_{k}}0\alpha\iota$be(resp.

belongsand$\omega 0\alpha_{t}$$\{\pi\psi(B)X\psi_{)}B\in B(\mathcal{H})\}$$(\alpha_{\mathrm{t}})_{\omega,\psi}$)with$B(\mathcal{H})$$\{\pi_{\omega}(A)x_{\omega}; A\in B(\mathcal{H})\}$$(\alpha_{t})_{\omega\psi}\pi\}\psi(B)x\psi=\pi_{\omega}((\alpha_{t})(B))x_{\omega}$

Then, for any $B,$

$C\in \mathcal{B}(\mathcal{H})$,$B\in B(\mathcal{H})$.

, we have$<(\alpha_{t})_{\omega,\psi}T\psi(B)x_{\psi 1(}\alpha t)_{\omega,\psi}T\psi(c)x_{\psi}>$$=$$<\pi_{\omega}((\alpha t)(B))x_{\omega}|\pi_{\omega}((\alpha_{t})(c))X_{\{}v>$$=$$<x_{\omega}|\pi_{\omega}((\alpha_{t})(B)^{*}(\alpha_{t})(c))X_{\omega}>$$=$$<x_{\omega}|\pi_{\omega}((\alpha_{t})(B^{*}C))_{X_{\{}>}v$$=\omega((\alpha_{t})(B^{*c}\mathrm{I})=\psi(B^{*}C)$$=$$<x_{\psi}|\pi_{\psi}(B^{*}c)x_{\psi}>$$=$$<\pi_{\psi}(B)X\psi|\pi_{\psi}(c)x\psi>$.

is the identity mapping. It is clear that theThese equalities imply thatis exactly equal to, because is surjecuniform closure of
can be uniquely extended to an isometry defined ontive. Therefore,. Since,$B,$, we havefor any$(\alpha_{t})_{\omega,\psi}^{*}(\alpha t)_{\omega},\emptyset$$\{\pi_{\omega}((\alpha_{t})(B))_{X_{\omega}}; B\in\beta(\mathcal{H})\}$$\mathcal{H}_{\omega}$$\alpha_{\mathrm{t}}$$(\alpha_{t})_{\omega,\psi}$$\mathcal{H}_{\psi}$$C\in B(\mathcal{H})$$(\alpha_{t})_{\omega},\psi\pi\psi(B)(\alpha t)_{(}*v,\psi^{\pi_{\omega}(}(\alpha_{t})(C))x_{\omega}$$=$$(\alpha_{t})_{\omega,\psi}\pi_{\psi}(B)(\alpha_{t})*\omega,\psi(\alpha t)_{(}\rho_{1}\psi\pi\psi(c)x_{\psi}$$=$$(\alpha_{t})_{\omega,\psi^{\pi\psi(BC}})X\psi$$=\pi_{\omega}((\alpha_{\mathrm{t}})(Bc))_{X_{\omega}}$$=\pi_{\{v}((\alpha_{\mathrm{t}})(B))T_{\omega}((\alpha t)(o))x_{\omega}$,34

these equalities imply thatand$(\alpha_{t})_{\omega,\psi}\pi_{\psi}(B)(\alpha \mathrm{t})_{\omega,\psi}^{*}=\pi_{\omega}((\alpha_{t})(B))$

holds for any$B\in B(\mathcal{H})$$\{T\psi((\alpha t)(B))x_{\psi;B}\in \mathcal{B}(\mathcal{H})\}’=(\alpha_{\mathrm{f}})_{\omega}^{*},\psi\{\pi(\omega(\alpha_{t})(B))X\omega;B\in B(\mathcal{H})\}’(\alpha t)_{\omega,\psi}--\mathbb{C}I$,,, and $A’$ means the commutant of an algewhere means the identity mapping on
bra . These equalities imply that the cyclic representation
is irreducible,
therefore, we obtain the conclusion.
$I$$\mathcal{H}\psi$$A$$\{\mathcal{H}_{\psi)}\pi_{\psi}, x_{\psi}\}$

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$C^{*}$-algebras and$W^{*}$-algebras, Springer Verlag, Berlin, 1971.