Analytic extension techniques for unitary representations of Banach-Lie groups

Let $(G,\theta )$ be a Banach–Lie group with involutive automorphism $\theta$, $\text{g}=\text{fh}\oplus \text{fq}$ be the $\theta$-eigenspaces in the Lie algebra $\text{g}$ of $G$, and $H=(G^{\theta }{)}_0$ be the identity component of its group of fixed points. An Olshanski semigroup is a semigroup $S\text{subeq}G$ of the form $S = H \exp(W)$,where$W$isanopen$\Ad(H)$-invariantconvexconein$\fq$ 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 $\pi S\to B(\text{cH})$ 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 ${\text{g}}_c=\text{fh}+i\text{fq}$. 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