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.

中文翻译：

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无极分解的*-子半群表示的解析扩展

令$(G,\tau)$ 是一个有限维李群，其对合自同构$\tau$ 为$G$，并令$\mathfrak g = \mathfrak h \oplus \mathfrak q $ 为其对应的李代数分解. 我们证明了在开 $*$-subsemigroup $S \subset G$ 的复 Hilbert 空间 $\mathcal H$ 上的每个非退化强连续表示，其中 $s^* = \tau(s)^{-1 }$，具有对李代数$[\mathfrak q,\mathfrak q] \oplus i\mathfrak q$ 的1-连通李群$G_1^c$ 的强连续酉表示的解析扩展。我们进一步研究了存在对李代数 $\mathfrak h \oplus i\mathfrak q$ 的 1-连通李群 $G^c$ 的解析扩展的最小条件。这个结果推广了 L\"uscher-Mack 定理和 L\" 的扩展 满足 $S = S(G^\tau)_0$ 的 $*$-子半群的 uscher-Mack 定理由 Merigon、Neeb 和 \'Olafsson 提出。最后，我们证明了某些 $*$-子半群 $S$ 的非退化强连续表示甚至可以扩展到 Olshanski 半群的广义版本的表示。