Extension of representations in quasi *-algebras

Abstract

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

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