Holomorphic geometric models for representations of C ∗-algebras

Representations of Banach–Lie groups are realized on Hilbert spaces formed by sections of holomorphic homogeneous vector bundles. These sections are obtained by means of a new notion of reproducing kernel, which is suitable for dealing with involutive diffeomorphisms defined on the base spaces of the bundles. The theory involves considering complexifications of homogeneous spaces acted on by groups of unitaries, and applies in particular to representations of C ∗-algebras endowed with conditional expectations. In this way, we present holomorphic geometric models for the Stinespring dilations of completely positive maps. The general results are further illustrated by a discussion of several specific topics, including similarity orbits of representations of amenable Banach algebras, similarity orbits of conditional expectations, geometric models of representations of Cuntz algebras, the relationship to endomorphisms of B(H), and non-commutative stochastic analysis