Extension of representations in quasi *-algebras

Abstract

Let $(\A, \A_o)$ be a topological \qxa, which means in particular that $\A_o$ is a topological \xa, dense in \A. Let $\pi^o$ be a *-representation of $\A_o$ in some pre-Hilbert space $\D \subset \H$. Then we present several ways of extending $\pi^o$, by closure, to some larger \qxa\ contained in \A, either by Hilbert space operators, or by sesquilinear forms on \D. Explicit examples are discussed, both abelian and nonabelian, including the CCR algebra.