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Let ( G , θ ) be a Banach–Lie group with involutive automorphism  θ ,be the  θ -eigenspaces in the Lie algebraof  G , and  H =( G   θ  ) 0  be the identity component of its group of fixed points. An Olshanski semigroup is a semigroup  S ⊆ G  of the form, where  W  is an open Ad( H )-invariant convex cone inand the polar mapis a diffeomorphism. Any such semigroup carries an involution * satisfying. Our central result, generalizing the Lüscher–Mack theorem for finite-dimensional groups, asserts that any locally bounded *-representationwith a dense set of smooth vectors defines by “analytic continuation” a unitary representation of the simply connected Lie group  G  c  with Lie algebra. 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.

Analytic Extension Techniques for Unitary Representations of Banach–Lie Groups