  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.
