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Finite dimensional semigroups of unitary endomorphisms of standard subspaces


Author: Karl-H. Neeb
Journal: Represent. Theory 25 (2021), 300-343
MSC (2020): Primary 22E45; Secondary 81R05, 81T05
DOI: https://doi.org/10.1090/ert/566
Published electronically: April 27, 2021
MathSciNet review: 4248936
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Abstract:

Let $\mathtt {V}$ be a standard subspace in the complex Hilbert space $\mathcal {H}$ and $G$ 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 \[ S_{\mathtt {V}} = \{ g\in G \cap \operatorname {U}(\mathcal {H})\colon g\mathtt {V} \subseteq \mathtt {V}\} \] 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 $G$. 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 $3$-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.


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Additional Information

Karl-H. Neeb
Affiliation: Department Mathematik, FAU Erlangen-Nürnberg, Cauerstrasse 11, 91058-Erlangen, Germany
MR Author ID: 288679
Email: neeb@math.fau.de

Received by editor(s): May 21, 2019
Received by editor(s) in revised form: June 29, 2020, and February 9, 2021
Published electronically: April 27, 2021
Additional Notes: The author was supported by DFG-grant NE 413/10-1.
Article copyright: © Copyright 2021 American Mathematical Society