Finite dimensional semigroups of unitary endomorphisms of standard subspaces,Representation Theory

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Finite dimensional semigroups of unitary endomorphisms of standard subspaces
Representation Theory ( IF 0.7 ) Pub Date : 2021-04-27 , DOI: 10.1090/ert/566
Karl-H. Neeb

Abstract:Let $\mathtt {V}$ be a standard subspace in the complex Hilbert space $\mathcal {H}$ and

be a finite dimensional Lie group of unitary and antiunitary operators on $\mathcal {H}$ containing the modular group $(\Delta _{\mathtt {V}}^{it})_{t \in \mathbb{R}}$ of $\mathtt {V}$ and the corresponding modular conjugation $J_{\mathtt {V}}$ . We study the semigroup

$\displaystyle S_{\mathtt {V}} = \{ g\in G \cap \operatorname {U}(\mathcal {H})\colon g\mathtt {V} \subseteq \mathtt {V}\}$

and determine its Lie wedge $\operatorname {\textbf {L}}(S_{\mathtt {V}}) = \{ x \in \mathfrak{g} \colon \exp (\mathbb{R}_+ x) \subseteq S_{\mathtt {V}}\}$ , i.e., the generators of its one-parameter subsemigroups in the Lie algebra $\mathfrak{g}$ of

. The semigroup $S_{\mathtt {V}}$ is analyzed in terms of antiunitary representations and their analytic extension to semigroups of the form $G \exp (iC)$ , where $C \subseteq \mathfrak{g}$ is an $\operatorname {Ad}(G)$ -invariant closed convex cone. Our main results assert that the Lie wedge $\operatorname {\textbf {L}}(S_{\mathtt {V}})$ spans a

-graded Lie subalgebra in which it can be described explicitly in terms of the involution $\tau$ of $\mathfrak{g}$ induced by $J_{\mathtt {V}}$ , the generator $h \in \mathfrak{g}^\tau$ of the modular group, and the positive cone of the corresponding representation. We also derive some global information on the semigroup $S_{\mathtt {V}}$ itself.

[Ar63]

中文翻译：

标准子空间unit同构的有限维半群

摘要：设是在复Hilbert空间标准的子空间和是一体和antiunitary运营商的有限维李群上含有模块化组的和相应的模块化缀合。我们研究半群 $\ mathtt {V}$ $\ mathcal {H}$

$\ mathcal {H}$ $（\ Delta _ {\ mathtt {V}} ^ {it}）_ {t \ in \ mathbb {R}}$ $\ mathtt {V}$ $J _ {\ mathtt {V}}$

$\ displaystyle S _ {\ mathtt {V}} = \ {g \ in G \ cap \ operatorname {U}（\ mathcal {H}）\冒号g \ mathtt {V} \ subseteq \ mathtt {V} \}$

并确定其李楔，即的李代数中一参数子半群的生成子。根据反unit表示及其对形式的半群的解析扩展来分析半群，其中-是不变的闭合凸锥。我们的主要结果断言烈楔跨度一个其中可以明确地在对合来描述-graded烈子代数的诱导，发电机模块化组和相应的表示的正锥形。我们还得出有关半群本身的一些全局信息。 $\ operatorname {\ textbf {L}}（S _ {\ mathtt {V}}）= \ {x \ in \ mathfrak {g} \冒号\ exp（\ mathbb {R} _ + x）\ subseteq S _ {\ mathtt {V}} \}$ $\ mathfrak {g}$

$S _ {\ mathtt {V}}$ $G \ exp（iC）$ $C \ subseteq \ mathfrak {g}$ $\ operatorname {Ad}（G）$ $\ operatorname {\ textbf {L}}（S _ {\ mathtt {V}}）$

$\ tau$ $\ mathfrak {g}$ $J _ {\ mathtt {V}}$ $h \ in \ mathfrak {g} ^ \ tau$ $S _ {\ mathtt {V}}$

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