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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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  • 1. F.Bagarello and C.Trapani, States and representations of $CQ^\ast$-algebras, Ann. Inst. H. Poincaré 61, 103-133 (1994) MR 95j:46062
  • 2. F.Bagarello and C.Trapani, CQ*-algebras: structure properties, Publ. Res. Inst. Math. Sci., Kyoto Univ. 32, 85-116 (1996) MR 97d:46064
  • 3. F.Bagarello and C.Trapani, $L^p$-spaces as quasi *-algebras, J. Math. Anal. Appl . 197, 810-824 (1996) MR 96k:46064
  • 4. F.Bagarello and C.Trapani, The Heisenberg dynamics of spin systems:a quasi $*$-algebras approach, J. Math. Phys. 37, 4219-4234 (1996) MR 97j:82005
  • 5. I.M.Gelfand and N.Ya.Vilenkin, Generalized functions Vol. 4, Academic Press, New York and London, 1964 MR 55:8786d
  • 6. G.Lassner, Algebras of unbounded operators and quantum dynamics, Physica, 124A, 471-479 (1984) CMP 17:01
  • 7. G.Lassner and G.A.Lassner, Qu$^*$-algebras and twisted product, Publ. Res. Inst. Math. Sci., Kyoto Univ. 25, 279-299 (1989) MR 90h:47087
  • 8. S.Stratila and L.Szido, Lectures on von Neumann Algebras, Revision of 1975 original, Abacus Press, Tunbridge Wells, 1979 MR 81j:46089
  • 9. M.Takesaki, Tomita's theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, 128 Springer-Verlag, 1970 MR 42:5061
  • 10. C.Trapani, Quasi $*$-algebras of operators and their applications, Rev. Math. Phys. 7 1303-1332 (1995) MR 97a:47070
  • 11. A.Van Daele, A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras, J. Functional Analysis 15, 378-393 (1974) MR 49:11264

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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

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