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Decomposition of operator semigroups on W\(^{\ast}\)-algebras. (English) Zbl 1250.47043

The Jacobs-DeLeeuw-Glicksberg decomposition of a Hilbert space \(H\) corresponding to a semigroup \(\mathcal{S}\) of contractions on \(H\) is a decomposition of \(H\) into \(\mathcal{S}\)-invariant subspaces: \(H=H_r \oplus H_s\), where \(H_r\) is the reversible and \(H_s\) the stable subspace, see [K. Jacobs, Math. Z. 67, 83–92 (1957; Zbl 0077.10901)], [K. de Leeuw and I. Glicksberg, Bull. Am. Math. Soc. 65, 134–139 (1959; Zbl 0089.32702); Acta Math. 105, 63–97 (1961; Zbl 0104.05501)]. The authors consider semigroups of operators on a \(W^*\)-algebra and prove, under appropriate assumptions, the existence of a Jacobs-de Leeuw-Glicksberg type decomposition in this case. As an application of the theory, the authors show a non-commutative version of the Perron-Frobenious theorem for \(W^*\)-algebras. The theory is also applied to study asymptotics of \(W^*\)-dynamical systems.

MSC:

47D03 Groups and semigroups of linear operators
46L45 Decomposition theory for \(C^*\)-algebras
37A55 Dynamical systems and the theory of \(C^*\)-algebras
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