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Zbl 0532.46040
Robinson, Derek W.
Strongly positive semigroups and faithful invariant states.
(English)
[J] Commun. Math. Phys. 85, 129-142 (1982). ISSN 0010-3616; ISSN 1432-0916/e

The results on noncommutative ergodic theory are proved in the following setting: M is a $W\sp*$-algebra, $\{\tau\sb t\vert t>0\}$ a semigroup of strongly positive (i.e. $\tau\sb t(A\sp*A)\ge \tau\sb t(A)\sp*\tau\sb t(A))$ linear maps of M into itself (no continuity assumptions of $\tau$ as a function of t is required), and $\omega$ is a faithful $\tau$- invariant normal state on M. It is shown that many results known in the case when $\tau$ is a group of *-automorphisms [{\it O. Bratteli} and the author, Operator algebras and quantum statistical mechanics, Vol. I (1979; Zbl 0421.46048)] can be extended to this situation. Some results of the paper [{\it A. Frigerio}, ibid. 63, 269-276 (1978; Zbl 0404.46050)] are also generalized. \par Among the results obtained in the paper are: \par i) a description of the set of invariant elements in M; \par ii) conditions that an invariant state $\omega$ have a unique decomposition into ergodic states; \par iii) a criterium of ergodicity of $\omega$ ; \par iv) in the case when $\tau$ is 2-positive, a strong positivity of a semigroup $\vert \tau \vert$ is proved, where $\vert \tau \vert$ is given by $\vert \tau\sb t\vert(A)\Omega =\vert T\sb t\vert A\Omega$ ($\Omega$ is the cyclic and separating vector associated with $\omega$ and $T\sb t$ sends $A\Omega$ into $\tau\sb t(A)\Omega$, $A\in M)$. It is shown that $\vert \tau \vert$-ergodicity of $\omega$ is equivalent to uniform clustering property with respect to $\tau$ : $\lim\sb{t\to \infty}\Vert \omega '{\bbfO}\tau\sb t-\omega \Vert =0$ for all normal states $\omega$ '.
[A.Lodkin]
MSC 2000:
*46L55 Noncommutative dynamical systems
46L40 Automorphisms of C*-algebras
46L30 States of C*-algebras

Keywords: strongly positive semigroups; semigroup of strongly positive linear maps; faithful invariant normal state; noncommutative ergodic theory; group of *-automorphisms; invariant elements; unique decomposition into ergodic states; cyclic and separating vector; uniform clustering property

Citations: Zbl 0421.46048; Zbl 0404.46050

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