Non-negative Spectral Measures and Representations of C*-Algebras

Regular normalized W-valued spectral measures on a compact Hausdorff space X are in one-to-one correspondence with unital *-representations ρ:C(X,C)→W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rho : C(X, \mathbb{C}) \to W}$$\end{document}, 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 W1, W2 there is a one-to-one correspondence between unital *-representations ρ:C(X,W1)→W2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rho : C(X, W_1) \to W_2}$$\end{document} and special B(W1, W2)-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 (Cimprič and Zalar, J Math Anal Appl 401:307–316, 2013) in connection with moment problems for operator polynomials.