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 \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.