Holomorphic geometric models for representations of $C^*$-algebras

TL;DR

This paper develops holomorphic geometric models for $C^*$-algebra representations using a novel reproducing kernel approach, with applications to dilation theory, similarity orbits, and non-commutative stochastic analysis.

Contribution

It introduces a new notion of reproducing kernel for holomorphic bundles, linking complex geometry with $C^*$-algebra representation theory and dilation techniques.

Findings

Reproducing kernel method effectively analyzes $C^*$-algebra representations.

Application to dilation theory of completely positive maps and Stinespring dilations.

Illustrations include models for Cuntz algebra representations and non-commutative stochastic processes.

Abstract

Representations of $C^*$-algebras are realized on section spaces of holomorphic homogeneous vector bundles. The corresponding section spaces are investigated by means of a new notion of reproducing kernel, suitable for dealing with involutive diffeomorphisms defined on the base spaces of the bundles. Applications of this technique to dilation theory of completely positive maps are explored and the critical role of complexified homogeneous spaces in connection with the Stinespring dilations is pointed out. 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 ${\mathcal B}({\mathcal H})$, and non-commutative stochastic analysis.