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Holomorphic representation theory. II. (English) Zbl 0842.22004
[Part I in Math. Ann. 301, 155-181 (1995; Zbl 0829.43017)].
A holomorphic representation of a complex Ol’shanskij semigroup \(S\) is a weakly continuous monoid morphism \(\pi : S \to B(H)\) into the algebra of bounded operators on a Hilbert space \(H\) such that \(\pi\) is holomorphic on the interior \(\text{int}(S)\) of \(S\) and \(\pi\) is involutive, i.e. \(\pi(s)^* = \pi(s)^*\) holds for all \(s\in S\). The author considers two principal problems of representation theory for this setting: (1) Describe the irreducible holomorphic representations of \(S\). (2) Decompose a holomorphic representation of \(S\) into irreducible representations.
A complete solution of (1) is obtained under the assumption that the group of inner automorphisms of \(g\) is closed in the group \(\text{Aut} (g)\) of all automorphisms of \(g\).
Reviewer: A.K.Guts (Omsk)

MSC:
22E15 General properties and structure of real Lie groups
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
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