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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, {τ t |t>0} a semigroup of strongly positive (i.e. τ t (A * A)τ t (A) * τ t (A)) linear maps of M into itself (no continuity assumptions of τ as a function of t is required), and ω is a faithful τ- invariant normal state on M. It is shown that many results known in the case when τ 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 ω have a unique decomposition into ergodic states;

iii) a criterium of ergodicity of ω ;

iv) in the case when τ is 2-positive, a strong positivity of a semigroup |τ| is proved, where |τ| is given by |τ t |(A)Ω=|T t |AΩ (Ω is the cyclic and separating vector associated with ω and T t sends AΩ into τ t (A)Ω, AM). It is shown that |τ|-ergodicity of ω is equivalent to uniform clustering property with respect to τ : lim t ω ' 𝕆τ t -ω=0 for all normal states ω ’.

Reviewer: A.Lodkin
MSC:
46L55Noncommutative dynamical systems
46L40Automorphisms of C * -algebras
46L30States 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