Topological partial *-algebras: Basic properties and examples

TL;DR

This paper explores the foundational properties of topological partial *-algebras, examining their structure, examples, and relation to other algebraic frameworks within functional analysis.

Contribution

It introduces the concept of topological partial *-algebras, analyzing their properties and providing examples from function spaces and operator algebras.

Findings

Characterization of topological partial *-algebras

Examples from L^p spaces and operator algebras

Connections to quasi *-algebras and CQ*-algebras

Abstract

Let $A$ be a partial *-algebra endowed with a topology $\tau$ that makes it into a locally convex topological vector space $A[\tau]$. Then $A$ is called a topological partial *-algebra if it satisfies a number of conditions, which all amount to require that the topology $\tau$ 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).