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 $\rho :C(X,\mathbb{C})\otimes W_1\to W_2$, 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$$\mathcal{B}$$standsforacommutative$\ast$-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$$\mathcal{L}$^+($\mathcal{D}$)$standsfora$\ast$-algebraofspeciallinearoperatorsonadensesubspace$$\mathcal{D}$$ofaHilbertspace$$\mathcal{K}$$.