Some classes of topological quasi $*$-algebras

Abstract

The completion A[τ] of a locally convex *-algebra A[τ] with not jointly continuous multiplication is a *-vector space with partial multiplication xy defined only for x or y ε A0, and it is called a topological quasi *-algebra. In this paper two classes of topological quasi *-algebras called strict CQ*-algebras and HCQ*-algebras are studied. Roughly speaking, a strict CQ*-algebra (resp. HCQ*-algebra) is a Banach (resp. Hubert) quasi *-algebra containing a C*-algebra endowed with another involution # and C*norm || ||#. HCQ*-algebras are closely related to left Hubert algebras. We shall show that a Hubert space is a HCQ*-algebra if and only if it contains a left Hubert algebra with unit as a dense subspace. Further, we shall give a necessary and sufficient condition under which a strict CQ*-aIgebra is embedded in a HCQ*-algebra. © 2001 American Mathematical Society.