Finite dimensional semigroups of unitary endomorphisms of standard subspaces

Abstract

Let $V$ be a standard subspace in the complex Hilbert space $H$ and $G$ be a finite dimensional Lie group of unitary and antiunitary operators on $H$ containing the modular group $(\Delta_V^{it})_{t \in R}$ of $V$ and the corresponding modular conjugation~$J_V$. We study the semigroup \[ S_V = \{ g\in G \cap U(H) : gV \subseteq V\} \] and determine its Lie wedge $L(S_V) = \{ x \in L(G) : exp(R_+ x) \subseteq S_V\}$, i.e., the generators of its one-parameter subsemigroups in the Lie algebra $L(G)$ of~$G$. The semigroup $S_V$ is analyzed in terms of antiunitary representations and their analytic extension to semigroups of the form $G exp(iC)$, where $C \subseteq L(G)$ is an $Ad(G)$-invariant closed convex cone. Our main results assert that the Lie wedge $L(S_V)$ spans a $3$-graded Lie subalgebra in which it can be described explicitly in terms of the involution $\tau$ of $L(G)$ induced by $J_V$, the generator $h \in L(G)^\tau$ of the modular group, and the positive cone of the corresponding representation. We also derive some global information on the semigroup $S_V$ itself