Standard systems for semifinite O-algebras

Author:
Atsushi Inoue

Journal:
Proc. Amer. Math. Soc. **125** (1997), 3303-3312

MSC (1991):
Primary 47D40; Secondary 46K15, 46L10

DOI:
https://doi.org/10.1090/S0002-9939-97-03962-2

MathSciNet review:
1403134

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We shall continue the study of standard systems which make it possible to develop the Tomita-Takesaki theory in O-algebras. The main purpose of this paper is to give the necessary and sufficient conditions for which a standard system of an O-algebra , a generalized vector and the commutant is unitarily equivalent to a standard system constructed by a standard tracial generalized vector for an O-algebra and a non-singular positive self-adjoint operator affiliated with the commutant of .

**[1]**J. -P. Antoine, A. Inoue, H. Ogi and C. Trapani,*Standard generalized vectors in the space of Hilbert-Schmidt operators*, Ann. Inst. Henri Poincaré**63**(1995), 177-210. MR**96i:47082****[2]**P. G. Dixon,*Unbounded operator algebras*, Proc. London Math. Soc.**23**(1971), 53-69. MR**45:911****[3]**S. P. Gudder and R. L. Scruggs,*Unbounded representations of -algebras*, Pacific J. Math.**70**(1977), 369-382. MR**58:2345****[4]**A. Inoue,*On a class of unbounded operator algebras I, II, III*, Pacific J. Math.**65**(1976), 77-95;**66**(1976), 411-431;**69**(1977), 105-115. MR**58:23641a**; MR**58:23641b**; MR**58:23641c****[5]**-*An unbounded generalization of the Tomita-Takesaki theory*, Publ. Res. Inst. Math. Soc., Kyoto Univ.**22**(1986), 725-765. MR**88c:47094****[6]**-*Standard generalized vectors for algebras of unbounded operators*, J. Math. Soc. Japan**47**(1995), 329-347. MR**96c:47065****[7]**A. Inoue and W. Karwowski,*Cyclic generalized vectors for algebras of unbounded operators*, Publ. Res. Inst. Math. Soc., Kyoto Univ.**30**(1994), 577-601. MR**96a:47081****[8]**G. Lassner,*Topological algebras of operators*, Rep. Math. Soc.**3**(1972), 279-293. MR**48:889****[9]**R. Pallu de la Barriére,*Algêbres unitaires et espaces d'Ambrose*, Ann. École Norm. Sup.**70**(1953), 381-401. MR**15:721e****[10]**R. T. Powers,*Self-adjoint algebras of unbounded operators*, Commun. Math. Phys.**21**(1971), 85-124. MR**44:811****[11]**-*Algebras of unbounded operators*, Proc. Sym. Pure Math.**38**(1982), 389-406. MR**84a:47059****[12]**M. A. Rieffel and A. Van Daele,*A bounded operator approach to Tomita-Takesaki theory*, Pacific J. Math.**69**(1977), 187-221. MR**55:11066****[13]**I. E. Segal,*A non-commutative extension of abstract integration*, Ann. Math.**57**(1953), 401-457. MR**14:991f**; MR**15:204h****[14]**K. Schmüdgen,*Unbounded Operator Algebras and Representation Theory*, Akademie-Verlag Berlin, 1990. MR**91f:47062****[15]**M. Takesaki,*Tomita's theory of modular Hilbert algebras and its applications*, Lecture Notes in Mathematics**128**Springer, 1970. MR**42:5061****[16]**A. Van Daele,*A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras*, J. Functional Analysis**15**(1974), 378-393. MR**49:11264**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
47D40,
46K15,
46L10

Retrieve articles in all journals with MSC (1991): 47D40, 46K15, 46L10

Additional Information

**Atsushi Inoue**

Affiliation:
Department of Applied Mathematics, Fukuoka University, Fukuoka, 814-80, Japan

Email:
sm010888ssat.fukuoka-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-97-03962-2

Keywords:
O$^*$-algebra,
standard generalized vector,
Tomita-Takesaki theory

Received by editor(s):
June 12, 1996

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1997
American Mathematical Society