J. -P. Antoine | Searcharxiv

arXiv2009

Extension of representations in quasi *-algebras

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

math-ph↗

arXiv2009

Topological partial *-algebras: Basic properties and examples

Let $A$ be a partial *-algebra endowed with a topology $τ$ that makes it into a locally convex topological vector space $A[τ]$. Then $A$ is called a topological partial *-algebra if it satisfies a number of conditions, which all amount to require that the topology $τ$ fits with the multiplier structure of $A$ Besides the obvious cases of topological quasi *-algebras and CQ*-algebras, we examine several classes of potential topological partial *-algebras, either function spaces (lattices of $L^p$ spaces on $[0,1]$ or on $\mathbb R$, amalgam spaces), or partial *-algebras of operators (operators on a partial inner product space, O*-algebras).

math-ph↗