Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
MathSciNet review: 1443873
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Abstract: A result of Haagerup, generalizing a theorem of Takesaki, states the following: If ${\mathcal{N}}\subset {\mathcal{M}}$ are von Neumann algebras, then there exists a faithful, normal and semi-finite (fns) operator valued weight $T \colon {\mathcal{M}}_{+} \rightarrow \widehat {{\mathcal{N}}_{+}}$ if and only if there exist fns weights $\tilde \varphi $ on ${\mathcal{M}}$ and $\varphi $ on ${\mathcal{N}}$ satisfying $\sigma ^{\varphi }_{t}(x) = \sigma ^{\tilde \varphi }_{t}(x) \, \forall x \in {\mathcal{N}}\ , t \in \mathbb{R}$. In fact, $T$ can be chosen such that $\tilde \varphi = \varphi \circ T$; $T$ is then uniquely determined by this condition. We present a proof of the above which does not use any structure theory.


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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: http://dx.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