On Well-behaved Unbounded Representations of *-Algebras

TL;DR

This paper introduces a framework for well-behaved unbounded *-representations of *-algebras using compatible pairs of normed *-algebras and their representations, with detailed examples illustrating the approach.

Contribution

It proposes a new general approach to unbounded *-representations of *-algebras via compatible pairs and establishes the existence of associated well-behaved representations.

Findings

Framework for unbounded *-representations using compatible pairs

Existence of associated well-behaved *-representations for any non-degenerate representation

Detailed examples demonstrating the applicability of the approach

Abstract

A general approach to the well-behaved unbounded *-representations of a *-algebra X is proposed. Let B be a normed *-algebra equipped with a left action |> of X on B such that (x |> a)^+ b=a^+(x^+ |> b) for a,b\in B and x\in X. Then the pair (X,B) is called a compatible pair. For any continuous non-degenerate *-representation \rho of B there exists a closed *-representation \rho' of X such that \rho'(x)\rho(b)=\rho(x |> b), where x\in X and b\in B. The *-representations \rho' are called the well-behaved *-representations associated with the compatible pair (X,B). A number of examples are developed in detail.