无极式分解的$*$ -子半群表示的解析扩张
Analytic extensions of representations of $*$-subsemigroups without polar decomposition
摘要：
设$(G,τ)$是一个有限维李群，其对合自同构$τ$是$G$的一个对合自同构，$\mathfrak g = \mathfrak h \oplus \mathfrak q $是其相应的李代数分解.证明了开$*$ -子半群$S \subset G$在复Hilbert空间$\mathcal H$上的每个非退化强连续表示都是1-连通李群$G_1^c$的强连续酉表示的解析扩张，其中$s^* = τ(s)^{-1}$是李代数$[\mathfrak q,\mathfrak q] \oplus i\mathfrak q$ .进一步研究了1-连通李群$G^c$与李代数$\mathfrak h \oplus i\mathfrak q$的解析扩张存在的极小条件。这一结果推广了Merigon，Neeb，and Schlafsson的Lüscher-Mack定理以及Lüscher-Mack定理对于满足$S = S(G^τ)_0$的$*$ -子半群的推广.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。
英文摘要：
Let $(G,τ)$ be a finite-dimensional Lie group with an involutive automorphism $τ$ 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 $s^* = τ(s)^{-1}$, 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 $*$-subsemigroups satisfying $S = S(G^τ)_0$ by Merigon, Neeb, and Ólafsson. 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.
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