Holomorphic representation theory. I (Q1345938)

The author introduces the concept of a complex Ol'shanskij semigroup \(S = \Gamma(g,W,\pi_1 (G))\). The main result (Sect. 3) consists of a characterization of those unitary representations of a connected Lie group \(G\) which can be extended to a holomorphic representation of an Ol'shanskij semigroup \(S = \Gamma(g,W, \pi_1 (G))\) with \(U(S)_0 \cong G\). A \(C^*\)-algebra \(C^*(S,\alpha)\) is defined for an Ol'shanskij semigroup \(S\) and an absolute value \(\alpha\) on \(S\) such that the representations of this \(C^*\)-algebra correspond to the \(\alpha\)- bounded holomorphic representation of \(S\).