Some classes of topological quasi \(*\)-algebras (Q2723495)

Topological quasi *-algebras were first introduced by Lassner for the mathematical description of some quantum physical models. Let \({\mathcal A}[\tau]\) be a locally convex *-algebra with not jointly continuous multiplication. The completion \({\overline {\mathcal A}}[\tau]\) of \({\mathcal A}[\tau]\) is a *-vector space with partial multiplication \(x y\) defined only for \(x\) or \(y \in {\mathcal A}_0\), which is called a topological quasi *-algebra. In this paper two classes of \({\overline {\mathcal A}}[\tau]\) so called strict \(CQ^*\)-algebras and \(HCQ^*\)-algebras are studied. It may say that a strict \(CQ^*\)-algebra (resp. \(HCQ^*\)-algebra) is a Banach (resp. Hilbert) quasi *-algebra containing a \(C^*\)-algebra endowed with another involution \(\#\) and \(C^*\)-norm \(\|\cdot \|_{\#}\). It is shown that a Hilbert space is a \(HCQ^*\)-algebra if and only if it contains a left Hilbert algebra with unit as a dense subspace. A necessary and sufficient condition under which a strict \(CQ^*\)-algebra is embedded in a \(HCQ^*\)-algebra is given.