# Analytic extensions of representations of $*$-subsemigroups without polar decomposition

## TL;DR

This paper extends representations of certain $*$-subsemigroups in Lie groups to analytic and unitary representations of associated Lie groups, generalizing classical theorems and introducing minimal conditions for such extensions.

## Contribution

It generalizes the L"uscher-Mack Theorem for $*$-subsemigroups without polar decomposition and explores conditions for analytic extensions to larger Lie groups.

## Key findings

- Every non-degenerate strongly continuous representation extends analytically to a unitary representation of $G_1^c$.
- Conditions for extension to $G^c$ are identified and minimal.
- Representations of certain $*$-subsemigroups can be extended to generalized Olshanski semigroups.

## 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 $*$-subsemigroup $S \subset G$, where $s^* = \tau(s)^{-1}$, 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 L\"uscher-Mack Theorem and the extensions of the L\"uscher-Mack Theorem for $*$-subsemigroups satisfying $S = S(G^\tau)_0$ by Merigon, Neeb, and \'Olafsson. Finally, we prove that non-degenerate strongly continuous representations of certain $*$-subsemigroups $S$ can even be extended to representations of a generalized version of an Olshanski semigroup.