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Some classes of topological quasi $*$-algebras

Authors: F. Bagarello, A. Inoue and C. Trapani
Journal: Proc. Amer. Math. Soc. 129 (2001), 2973-2980
MSC (2000): Primary 46K70
Published electronically: March 14, 2001
MathSciNet review: 1840102
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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 $\text{{\scriptsize$\char93 $ }}$ and C$^*$-norm $\Vert \Vert _{\text{{\scriptsize$\char93 $ }}}$. HCQ$^*$-algebras are closely related to left Hilbert algebras. We shall show that a Hilbert space is a HCQ$^*$-algebra if and only if it contains a left Hilbert algebra with unit as a dense subspace. Further, we shall give a necessary and sufficient condition under which a strict CQ$^*$-algebra is embedded in a HCQ$^*$-algebra.

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Additional Information

F. Bagarello
Affiliation: Dipartimento di Matematica, Università di Palermo, I-90128 Palermo, Italy

A. Inoue
Affiliation: Department of Applied Mathematics, Fukuoka University, J-814-80 Fukuoka, Japan

C. Trapani
Affiliation: Dipartimento di Scienze Fisiche ed Astronomiche, Università di Palermo, I-90123 Palermo, Italy

Keywords: Topological quasi $*$-algebras, CQ$^*$-algebras, HCQ$^*$-algebras
Received by editor(s): February 20, 2000
Published electronically: March 14, 2001
Communicated by: David R. Larson
Article copyright: © Copyright 2001 American Mathematical Society