Operator valued weights without structure theory
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- by Tony Falcone and Masamichi Takesaki PDF
- Trans. Amer. Math. Soc. 351 (1999), 323-341 Request permission
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.References
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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
- MR Author ID: 170305
- Email: mt@math.ucla.edu
- Received by editor(s): January 30, 1997
- Additional Notes: This work is supported, in part, by NSF Grant DMS95-00882.
- © Copyright 1999 American Mathematical Society
- 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