Some classes of topological quasi *-algebras

Bagarello, F.; Inoue, A.; Trapani, C.

doi:10.1090/S0002-9939-01-06019-1

Some classes of topological quasi $*$-algebras
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by F. Bagarello, A. Inoue and C. Trapani PDF

Proc. Amer. Math. Soc. 129 (2001), 2973-2980 Request permission

Abstract:

The completion $\overline {\mathcal A}[\tau ]$ of a locally convex $*$-algebra $\mathcal A[\tau ]$ with not jointly continuous multiplication is a $*$-vector space with partial multiplication $xy$ defined only for $x$ or $y \in {\mathcal A}_{0}$, 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. Hilbert) quasi $*$-algebra containing a C$^*$-algebra endowed with another involution # and C$^*$-norm $\| \ \|_{\#}$. HCQ$^*$-algebras are closely related to left Hilbert algebras. We shall show that a Hilbert space is a HCQ$^*$-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$^*$-algebra is embedded in a HCQ$^*$-algebra.

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