Integral representations of *-representations of *-algebras

Regular normalized $B(W_1,W_2)$-valued non-negative spectral measures introduced in \cite{Zalar2014} are in one-to-one correspondence with unital $\ast$-representations $\rho:C(X,\mathbb{C})\otimes W_1 \rightarrow W_2$, where $X$ stands for a compact Hausdorff space and $W_1, W_2$ stand for von Neumann algebras. In this paper we generalize this result in two directions. The first is to $\ast$-representations of the form $\rho:\mathcal{B}\otimes W_1\rightarrow W_2$, where $\mathcal{B}$ stands for a commutative $\ast$-algebra $\mathcal{B}$, and the second is to special (not necessarily bounded) $\ast$-representations of the form $\rho:\mathcal{B}\otimes W_1\rightarrow \mathcal{L}^+(\mathcal{D})$, where $\mathcal{L}^+(\mathcal{D})$ stands for a $\ast$-algebra of special linear operators on a dense subspace $\mathcal{D}$ of a Hilbert space $\mathcal{K}$.