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Strongly positive semigroups and faithful invariant states. (English) Zbl 0532.46040
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. Among the results obtained in the paper are: i) a description of the set of invariant elements in M; ii) conditions that an invariant state $\omega$ have a unique decomposition into ergodic states; iii) a criterium of ergodicity of $\omega$ ; 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 $\omega$ ’.
Reviewer: A.Lodkin

46L55Noncommutative dynamical systems
46L40Automorphisms of $C^*$-algebras
46L30States of $C^*$-algebras
Full Text: DOI
[1] Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics, Vol. I. Berlin, Heidelberg, New York: Springer 1979 · Zbl 0421.46048
[2] Frigerio, A.: Stationary states of quantum dynamical semigroups. Commun. Math. Phys.63, 269--276 (1978) · Zbl 0404.46050 · doi:10.1007/BF01196936
[3] Evans, D.E.: Irreducible quantum dynamical semigroups. Commun. Math. Phys.54, 293--297 (1977) · Zbl 0374.46051 · doi:10.1007/BF01614091
[4] Bratteli, O., Robinson, D.W.: Unbounded derivations of von Neumann algebras. Ann. Inst. H. Poincaré25 (A), 139--164 (1976) · Zbl 0332.46043
[5] Majewski, A., Robinson, D.W.: Strictly positive and strongly positive semigroups. University of New South Wales Preprint (to be published in the Australian Journal of Mathematics) · Zbl 0521.47022
[6] Radin, C.: Non-commutative mean ergodic theory. Commun. Math. Phys.21, 291--302 (1971) · Zbl 0211.43504 · doi:10.1007/BF01645751
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[8] Choi, M.-D.: Inequalities for positive linear maps. J. Operat. Theory4, 271--285 (1980) · Zbl 0511.46051
[9] Davies, E.B.: Irreversible dynamics of infinite fermion systems. Commun. Math. Phys.55, 231--258 (1977) · Zbl 0361.47013 · doi:10.1007/BF01614549