Finite dimensional semigroups of unitary endomorphisms of standard subspaces

Neeb, Karl-H.

doi:10.1090/ert/566

Finite dimensional semigroups of unitary endomorphisms of standard subspaces
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by Karl-H. Neeb

Represent. Theory 25 (2021), 300-343

DOI: https://doi.org/10.1090/ert/566

Published electronically: April 27, 2021

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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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