Locally convex quasi ⁎-algebras with sufficiently many ⁎-representations

The main aim of this paper is the investigation of conditions under which a locally convex quasi ⁎-algebra ( A [ τ ] , A 0 ) attains sufficiently many ( τ , t w ) -continuous ⁎-representations in L † ( D , H ) , to separate its points. Having achieved this, a usual notion of bounded elements on A [ τ ] rises. On the other hand, a natural order exists on ( A [ τ ] , A 0 ) related to the topology τ, that also leads to a kind of bounded elements, which we call “order bounded”. What is important is that under certain conditions the latter notion of boundedness coincides with the usual one. Several nice properties of order bounded elements are extracted that enrich the structure of locally convex quasi ⁎-algebras.