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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 ${W}^{*}$-algebra, $\left\{{\tau }_{t}|t>0\right\}$ a semigroup of strongly positive (i.e. ${\tau }_{t}\left({A}^{*}A\right)\ge {\tau }_{t}{\left(A\right)}^{*}{\tau }_{t}\left(A\right)\right)$ 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 [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 [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 $|\tau |$ is proved, where $|\tau |$ is given by $|{\tau }_{t}|\left(A\right){\Omega }=|{T}_{t}|A{\Omega }$ (${\Omega }$ is the cyclic and separating vector associated with $\omega$ and ${T}_{t}$ sends $A{\Omega }$ into ${\tau }_{t}\left(A\right){\Omega }$, $A\in M\right)$. It is shown that $|\tau |$-ergodicity of $\omega$ is equivalent to uniform clustering property with respect to $\tau$ : ${lim}_{t\to \infty }\parallel {\omega }^{\text{'}}𝕆{\tau }_{t}-\omega \parallel =0$ for all normal states $\omega$ ’.

Reviewer: A.Lodkin
##### MSC:
 46L55 Noncommutative dynamical systems 46L40 Automorphisms of ${C}^{*}$-algebras 46L30 States of ${C}^{*}$-algebras
##### References:
 [1] Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics, Vol. I. Berlin, Heidelberg, New York: Springer 1979 [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) [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) [6] Radin, C.: Non-commutative mean ergodic theory. Commun. Math. Phys.21, 291–302 (1971) · Zbl 0211.43504 · doi:10.1007/BF01645751 [7] Choi, M.-D.: A Schwarz inequality for positive linear maps onC*-algebras. Ill. J. Math.18, 565–574 (1974) [8] Choi, M.-D.: Inequalities for positive linear maps. J. Operat. Theory4, 271–285 (1980) [9] Davies, E.B.: Irreversible dynamics of infinite fermion systems. Commun. Math. Phys.55, 231–258 (1977) · Zbl 0361.47013 · doi:10.1007/BF01614549