Strongly positive semigroups and faithful invariant states
type: Item
instance of: scholarly article
title: Strongly positive semigroups and faithful invariant states
zbMATH Open document ID: 0532.46040
DOI: 10.1007/BF02029138
author: Derek W. Robinson
published in: Communications in Mathematical Physics
publication date: 1982-01-01T00:00:00Z
review text: The results on noncommutative ergodic theory are proved in the following setting: M is a \(W^*\)-algebra, \(\{\tau_ t| t>0\}\) a semigroup of strongly positive (i.e. \(\tau_ t(A^*A)\geq \tau_ t(A)^*\tau_ t(A))\) 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 [\textit{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 [\textit{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 \(| \tau_ t|(A)\Omega =| T_ t| A\Omega\) (\(\Omega\) is the cyclic and separating vector associated with \(\omega\) and \(T_ t\) sends \(A\Omega\) into \(\tau_ t(A)\Omega\), \(A\in M)\). It is shown that \(| \tau |\)-ergodicity of \(\omega\) is equivalent to uniform clustering property with respect to \(\tau\) : \(\lim_{t\to \infty}\| \omega '{\mathbb{O}}\tau_ t-\omega \| =0\) for all normal states \(\omega\) '.
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L40
Mathematics Subject Classification ID: 46L30
zbMATH DE Number: 3844410
zbMATH Keywords: strongly positive semigroups
zbMATH Keywords: semigroup of strongly positive linear maps
zbMATH Keywords: faithful invariant normal state
zbMATH Keywords: noncommutative ergodic theory
zbMATH Keywords: group of *-automorphisms
zbMATH Keywords: invariant elements
zbMATH Keywords: unique decomposition into ergodic states
zbMATH Keywords: cyclic and separating vector
zbMATH Keywords: uniform clustering property
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3844410