Operator valued weights

without structure theory

Authors:
Tony Falcone and Masamichi Takesaki

Journal:
Trans. Amer. Math. Soc. **351** (1999), 323-341

MSC (1991):
Primary 46L50; Secondary 22D25

DOI:
https://doi.org/10.1090/S0002-9947-99-02028-0

MathSciNet review:
1443873

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Abstract | References | Similar Articles | Additional Information

Abstract: A result of Haagerup, generalizing a theorem of Takesaki, states the following: If are von Neumann algebras, then there exists a faithful, normal and semi-finite (fns) operator valued weight if and only if there exist fns weights on and on satisfying . In fact, can be chosen such that ; is then uniquely determined by this condition. We present a proof of the above which does not use any structure theory.

**[Haa1]**U. Haagerup,*Operator Valued Weights in von Neumann Algebras, I*, J. Func. Anal.**32**175-206 (1979). MR**81e:46049a****[Haa2]**U. Haagerup,*Operator Valued Weights in von Neumann Algebras, II*, J. Func. Anal.**33**339-361 (1979). MR**81e:46049b****[Hir]**M. Hirakawa,*A Generalization of -conditional Expectation and Operator Valued Weight*, Publ. Res. Inst. Math. Sci.**28**289-297 (1992). MR**93a:46121****[Mas]**T. Masuda,*A Note on a Theorem of A. Connes on Radon-Nikodym Cocycles*, Publ. Res. Inst. Math. Sci.**20**131-136 (1984). MR**85g:46081****[Tak1]**M. Takesaki,*Conditional Expectations in von Neumann Algebras*, J. Func. Anal.,**9**306-320 (1972). MR**46:2445****[Tak2]**M. Takesaki,*Theory of Operator Algebras I*, Springer-Verlag, Chapter III §2 (1979). MR**81e:46038****[Tak3]**M. Takesaki,*Theory of Operator Algebras II*, Chapter VIII §3; to appear.

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Additional Information

**Tony Falcone**

Affiliation:
Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1555

Address at time of publication:
Department of Mathematics, Illinois State University, Normal, Illinois 61790-4520

Email:
afalcone@math.ilstu.edu

**Masamichi Takesaki**

Affiliation:
Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1555

Email:
mt@math.ucla.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02028-0

Received by editor(s):
January 30, 1997

Additional Notes:
This work is supported, in part, by NSF Grant DMS95-00882.

Article copyright:
© Copyright 1999
American Mathematical Society