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

DOI:
https://doi.org/10.1090/S0002-9939-01-06019-1

Published electronically:
March 14, 2001

MathSciNet review:
1840102

Full-text PDF

Abstract | References | Similar Articles | Additional Information

The completion of a locally convex -algebra with not jointly continuous multiplication is a -vector space with partial multiplication defined only for or , 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.

**1.**F.Bagarello and C.Trapani,*States and representations of -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,*-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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46K70

Retrieve articles in all journals with MSC (2000): 46K70

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

DOI:
https://doi.org/10.1090/S0002-9939-01-06019-1

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