Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Relative modular theory for a weight

Author(s): Hideaki Izumi
Journal: Proc. Amer. Math. Soc. 127 (1999), 2703-2713.
MSC (1991): Primary 46L10
Posted: April 15, 1999
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We consider the balanced weight $\chi$ of a semi-finite weight $\varphi$ and a (not necessarily faithful) normal positive functional $\psi$ on a von Neumann algebra $\mathcal M$, and discuss how the modular operator $\Delta _\chi$ and the modular conjugation $J_\chi$ are described under the identification of the standard Hilbert space $\mathcal{H}_\chi$ with $\mathcal{H}_\varphi \oplus p\mathcal{H}_\varphi\oplus p'\mathcal{H}_\varphi\oplus pp'\mathcal{H}_\varphi$, where $p$ is the support projection of $\psi$ and $p'=J_\varphi p J_\varphi\in\mathcal{M}'$.

References:

1.
H. Araki, Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon-Nikodym theorem with a chain rule., Pacific J. Math. 50 (1974), 309-354. MR 50:2929

2.
A. Connes, Caractérisation des espaces vectoriels ordonnés sous-jacent aux algèbres de von Neumann, Ann. Inst. Fourier 24 (1974), 121-155. MR 51:13705

3.
T. Digernes, Duality for weights on covariant systems and its applications, Dissertation, Los Angeles, 1975.

4.
U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975), 271-283. MR 53:11387

5.
H. Izumi, Natural bilinear forms, natural sesquilinear forms and the associated duality of non-commutative $L^p$-spaces, to appear in Int. J. Math.

6.
H. Kosaki, Canonical $L^p$-Spaces Associated with an Arbitrary Abstract von Neumann Algebra, Thesis, UCLA, 1980.

7.
H. Kosaki, Applications of the Complex Interpolation Method to a von Neumann Algebra: Non-commutative $L^p$-Spaces, J. Funct. Anal. 56 (1984), 29-78. MR 86a:46085

8.
S. Str\u{a}til\u{a}, Modular Theory in Operator Algebras, Abacus Press, 1981. MR 85g:46072

9.
M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, 1979. MR 81e:46038

10.
M. Takesaki, The structure of operator algebras (in Japanese), Iwanami Shoten, 1983. MR 86f:46062

11.
M. Takesaki, Theory of Operator Algebras II (to be published).

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46L10

Retrieve articles in all Journals with MSC (1991): 46L10

Additional Information:

Hideaki Izumi
Affiliation: Mathematical Institute, Tohoku University, Aza-Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan
Email: h-izumi@math.tohoku.ac.jp

DOI: 10.1090/S0002-9939-99-04840-6
PII: S 0002-9939(99)04840-6
Received by editor(s): March 31, 1997
Received by editor(s) in revised form: November 24, 1997
Posted: April 15, 1999
Communicated by: David R. Larson
Copyright of article: Copyright 1999, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google