# Holomorphic geometric models for representations of $C^*$-algebras - Mathematics > Operator Algebras

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Abstract: Representations of $C^*$-algebras are realized on section spaces ofholomorphic homogeneous vector bundles. The corresponding section spaces areinvestigated by means of a new notion of reproducing kernel, suitable fordealing with involutive diffeomorphisms defined on the base spaces of thebundles. Applications of this technique to dilation theory of completelypositive maps are explored and the critical role of complexified homogeneousspaces in connection with the Stinespring dilations is pointed out. The generalresults 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 ofrepresentations of Cuntz algebras, the relationship to endomorphisms of${\mathcal B}{\mathcal H}$, and non-commutative stochastic analysis.

Author: Daniel Beltita, Jose E. Gale

Source: https://arxiv.org/