Non-negative spectral measures and representations of C*-algebras

TL;DR

This paper establishes a correspondence between unital *-representations of certain C*-algebras and non-negative spectral measures, extending spectral measure theory to operator-valued contexts.

Contribution

It introduces non-negative spectral measures for operator-valued C*-algebras and proves their one-to-one correspondence with unital *-representations, generalizing classical spectral measure theory.

Findings

Established a bijective correspondence between *-representations and non-negative spectral measures.

Extended spectral measure theory to operator-valued C*-algebras.

Connected spectral measures with moment problems for operator polynomials.

Abstract

Regular normalized W-valued spectral measures on a compact Hausdorff space X are in one-to-one correspondence with unital *-representations \rho:C(X)\to W, where W stands for a von Neumann algebra. In this paper we show that for every compact Hausdorff space X and every von Neumann algebras W_1,W_2 there is a one-to-one correspondence between unital *-representations \rho:C(X,W_1)\to W_2 and special B(W_1,W_2)-valued measures on X that we call non-negative spectral measures. Such measures are special cases of non-negative measures that we introduced in our previous paper in connection with moment problems for operator polynomials.