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.References
- F. Bagarello and C. Trapani, States and representations of $CQ^\ast$-algebras, Ann. Inst. H. Poincaré Phys. Théor. 61 (1994), no. 1, 103–133 (English, with English and French summaries). MR 1303188
- Fabio Bagarello and Camillo Trapani, $CQ^*$-algebras: structure properties, Publ. Res. Inst. Math. Sci. 32 (1996), no. 1, 85–116. MR 1384752, DOI 10.2977/prims/1195163181
- F. Bagarello and C. Trapani, $L^p$-spaces as quasi $^*$-algebras, J. Math. Anal. Appl. 197 (1996), no. 3, 810–824. MR 1373082, DOI 10.1006/jmaa.1996.0055
- F. Bagarello and C. Trapani, The Heisenberg dynamics of spin systems: a quasi$^*$-algebras approach, J. Math. Phys. 37 (1996), no. 9, 4219–4234. MR 1408088, DOI 10.1063/1.531652
- I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
- G.Lassner, Algebras of unbounded operators and quantum dynamics, Physica, 124A, 471-479 (1984)
- Gerd Lassner and Gisela A. Lassner, $Qu^*$-algebras and twisted product, Publ. Res. Inst. Math. Sci. 25 (1989), no. 2, 279–299. MR 1003789, DOI 10.2977/prims/1195173612
- Şerban Strătilă and László Zsidó, Lectures on von Neumann algebras, Editura Academiei, Bucharest; Abacus Press, Tunbridge Wells, 1979. Revision of the 1975 original; Translated from the Romanian by Silviu Teleman. MR 526399
- M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, Vol. 128, Springer-Verlag, Berlin-New York, 1970. MR 0270168
- C. Trapani, Quasi $^*$-algebras of operators and their applications, Rev. Math. Phys. 7 (1995), no. 8, 1303–1332. MR 1369745, DOI 10.1142/S0129055X95000475
- Alfons van Daele, A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras, J. Functional Analysis 15 (1974), 378–393. MR 0346539, DOI 10.1016/0022-1236(74)90029-9
Additional Information
- F. Bagarello
- Affiliation: Dipartimento di Matematica, Università di Palermo, I-90128 Palermo, Italy
- Email: bagarello@www.unipa.it
- A. Inoue
- Affiliation: Department of Applied Mathematics, Fukuoka University, J-814-80 Fukuoka, Japan
- Email: a-inoue@fukuoka-u.ac.jp
- C. Trapani
- Affiliation: Dipartimento di Scienze Fisiche ed Astronomiche, Università di Palermo, I-90123 Palermo, Italy
- Email: trapani@unipa.it
- Received by editor(s): February 20, 2000
- Published electronically: March 14, 2001
- Communicated by: David R. Larson
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2973-2980
- MSC (2000): Primary 46K70
- DOI: https://doi.org/10.1090/S0002-9939-01-06019-1
- MathSciNet review: 1840102