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

Abstract

Regular normalized W-valued spectral measures on a compact Hausdorff space X are in one-to-one correspondence with unital *-representations ${\rho : C(X, \mathbb{C}) \to W}$, 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 (Cimprič and Zalar, J Math Anal Appl 401:307–316, 2013) in connection with moment problems for operator polynomials.