  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 $\mathcal{K}$.
