Integral representations of *-representations of *-algebras

Abstract

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 $\rho :\mathcal{B}\otimes W_1\to W_2$, 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 $\rho :\mathcal{B}\otimes W_1\to {\mathcal{L}}^{+}(\mathcal{D})$, where ${\mathcal{L}}^{+}(\mathcal{D})$ stands for a $\ast$-algebra of special linear operators on a dense subspace $\mathcal{D}$ of a Hilbert space $\mathcal{K}$.

22 pages