Topological partial *-algebras: Basic properties and examples

Let Θ be a partial *-algebra endowed with a topology τ that makes it into a locally convex topological vector space Θ[τ]. Then Θ is called a topological partial *-algebra if it satisfies a number of conditions, which all amount to require that the topology T fits with the multiplier structure of Θ. 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 Lp spaces on [0,1] or on R, amalgam spaces), or partial *-algebras of operators (operators on a partial inner product space, O*-algebras).