Some classes of topological quasi *-algebras

Abstract

The completion $\overline{A}[\tau ]$ of a locally convex *-algebra $A[\tau ]$ with not jointly continuous multiplication is a *-vector space with partial multiplication $xy$ defined only for $x$ or $y\in A_0$, and it is called a topological quasi *-algebra. In this paper two classes of topological quasi *-algebras called strict CQ${}^{\ast }$-algebras and HCQ${}^{\ast }$-algebras are studied. Roughly speaking, a strict CQ${}^{\ast }$-algebra (resp. HCQ${}^{\ast }$-algebra) is a Banach (resp. Hilbert) quasi *-algebra containing a C${}^{\ast }$-algebra endowed with another involution $♯$ and C${}^{\ast }$-norm $\Vert {\Vert }_{♯}$. HCQ${}^{\ast }$-algebras are closely related to left Hilbert algebras. We shall show that a Hilbert space is a HCQ${}^{\ast }$-algebra if and only if it contains a left Hilbert algebra with unit as a dense subspace. Further, we shall give a necessary and sufficient condition under which a strict CQ${}^{\ast }$-algebra is embedded in a HCQ${}^{\ast }$-algebra.