Integral representations of *-representations of *-algebras

TL;DR

This paper extends the integral representation theory of *-representations of *-algebras, including unbounded cases, establishing a correspondence with spectral measures for broader classes of algebraic structures.

Contribution

It generalizes previous results to *-representations of the form W_1W_2, including unbounded operators, broadening the scope of integral representation theory.

Findings

Established a one-to-one correspondence with spectral measures for new classes of *-representations.

Extended integral representation results to unbounded *-representations.

Generalized the framework to include *-representations on dense subspaces of Hilbert spaces.

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 $\rho:C(X,\mathbb{C})\otimes W_1 \rightarrow W_2$, 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 $\mathcal{K}$.