# Analytic Extensions of Representations of *-Subsemigroups Without Polar Decomposition

Oeh D (2021)

**Publication Type:** Journal article

**Publication year:** 2021

### Journal

**Book Volume:** 2021

**Pages Range:** 6200-6245

**Journal Issue:** 8

**DOI:** 10.1093/imrn/rnz342

### Abstract

Let (G, tau) be a finite-dimensional Lie group with an involutive automorphism tau of G and let g = h circle plus q be its corresponding Lie algebra decomposition. We show that every nondegenerate strongly continuous representation on a complex Hilbert space 7-t of an open *-subsemigroup S subset of 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 [q, q] circle plus iq. We further examine the minimal conditions under which an analytic extension to the 1-connected Lie group G(c) with Lie algebra h circle plus iq exists. This result generalizes the Lascher-Mack theorem and the extensions of the Lascher-Mack theorem for *subsemigroups satisfying S = S(G(tau))(0) by Merigon, Neeb, and Olafsson. Finally, we prove that nondegenerate strongly continuous representations of certain *-subsemigroups S can even be extended to representations of a generalized version of an Olshanski semigroup.

### How to cite

**APA:**

Oeh, D. (2021). Analytic Extensions of Representations of *-Subsemigroups Without Polar Decomposition. *International Mathematics Research Notices*, *2021*(8), 6200-6245. https://doi.org/10.1093/imrn/rnz342

**MLA:**

Oeh, Daniel. "Analytic Extensions of Representations of *-Subsemigroups Without Polar Decomposition." *International Mathematics Research Notices* 2021.8 (2021): 6200-6245.

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