Analytic extensions of representations of $\ast$-subsemigroups without polar decomposition

Abstract

Let $(G,\tau )$ be a finite-dimensional Lie group with an involutive automorphism $\tau$ of $G$ and let $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{q}$ be its corresponding Lie algebra decomposition. We show that every non-degenerate strongly continuous representation on a complex Hilbert space $\mathcal{H}$ of an open $\ast$-subsemigroup $S\subset G$, where $s^{\ast }=\tau (s{)}^{-1}$, has an analytic extension to a strongly continuous unitary representation of the 1-connected Lie group $G_1^c$ with Lie algebra $[\mathfrak{q},\mathfrak{q}]\oplus i\mathfrak{q}$. We further examine the minimal conditions under which an analytic extension to the 1-connected Lie group $G^c$ with Lie algebra $\mathfrak{h}\oplus i\mathfrak{q}$ exists. This result generalizes the Lüscher-Mack Theorem and the extensions of the Lüscher-Mack Theorem for $\ast$-subsemigroups satisfying $S=S(G^{\tau }{)}_0$ by Merigon, Neeb, and Ólafsson. Finally, we prove that non-degenerate strongly continuous representations of certain $\ast$-subsemigroups $S$ can even be extended to representations of a generalized version of an Olshanski semigroup.

arXiv commentSection 3.3: Added more information about conditions for main theorem; Corrected typos