Analytic extension techniques for unitary representations of Banach-Lie groups

Abstract

Let $(G,\theta)$ be a Banach--Lie group with involutive automorphism $\theta$, $\g = \fh \oplus \fq$ be the $\theta$-eigenspaces in the Lie algebra $\g$ of $G$, and $H = (G^\theta)_0$ be the identity component of its group of fixed points. An Olshanski semigroup is a semigroup $S \subeq G$ of the form $S = H \exp(W)$, where $W$ is an open $\Ad(H)$-invariant convex cone in $\fq$ and the polar map $H \times W \to S, (h,x) \mapsto h \exp x$ is a diffeomorphism. Any such semigroup carries an involution * satisfying $(h\exp x)^* = (\exp x) h^{-1}$. Our central result, generalizing the L\"uscher--Mack Theorem for finite dimensional groups, asserts that any locally bounded *-representation $\pi \: S \to B(\cH)$ with a dense set of smooth vectors defines by "analytic continuation" a unitary representation of the simply connected Lie group $G_c$ with Lie algebra $ \g_c = \fh + i \fq$. We also characterize those unitary representations of $G_c$ obtained by this construction. With similar methods, we further show that semibounded unitary representations extend to holomorphic representations of complex Olshanski semigroups