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On semigroups of linear operators. (English) Zbl 1084.46529

Summary: In this note we discuss some groups of linear operators on \(C^*\)-algebras and their infinitesimal generators. We show that if \(\delta\) is the infinitesimal generator of a strongly countinuous one-parameter group of automorphisms \(\{\alpha_t\}_{t\in\mathbb{R}}\) such that \(\delta\) is approximately inner in a core for \(\delta\), then \(\{\alpha_t\}_{t\in\mathbb{R}}\) is approximately inner. We also prove that the infinitesimal generator of some strongly continuous automorphism groups of \(C^*\)-subalgebras of operators is strongly approximately inner on some dense subset of a Hilbert space.

MSC:

46L05 General theory of \(C^*\)-algebras
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
47D03 Groups and semigroups of linear operators
47D06 One-parameter semigroups and linear evolution equations
47L45 Dual algebras; weakly closed singly generated operator algebras
47B47 Commutators, derivations, elementary operators, etc.
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