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Ueda's peak set theorem for general von Neumann algebras


Authors: David P. Blecher and Louis Labuschagne
Journal: Trans. Amer. Math. Soc. 370 (2018), 8215-8236
MSC (2010): Primary 46L51, 46L52, 47L75, 47L80; Secondary 03E10, 03E35, 03E55, 46J15, 46K50, 47L45
DOI: https://doi.org/10.1090/tran/7275
Published electronically: August 21, 2018
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Abstract: We extend Ueda's peak set theorem for subdiagonal subalgebras of tracial finite von Neumann algebras to $ \sigma $-finite von Neumann algebras (that is, von Neumann algebras with a faithful state, which includes those on a separable Hilbert space or with separable predual). To achieve this extension, completely new strategies had to be invented at certain key points, ultimately resulting in a more operator algebraic proof of the result. Ueda showed in the case of finite von Neumann algebras that his peak set theorem is the fountainhead of many other very elegant results, like the uniqueness of the predual of such subalgebras, a highly refined F & M Riesz type theorem, and a Gleason-Whitney theorem. The same is true in our more general setting, and indeed we obtain a quite strong variant of the last mentioned theorem. We also show that set theoretic issues dash hopes for extending the theorem to some other large general classes of von Neumann algebras, for example finite or semi-finite ones. Indeed certain cases of Ueda's peak set theorem for a von Neumann algebra $ M$ may be seen as `set theoretic statements' about $ M$ that require the sets to not be `too large'.


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Additional Information

David P. Blecher
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: dblecher@math.uh.edu

Louis Labuschagne
Affiliation: DST-NRF CoE in Mathematical and Statistical Sciences, Unit for BMI, Internal Box 209, School of Computer, Statistical, and Mathematical Sciences, North-West University, PVT. BAG X6001, 2520 Potchefstroom, South Africa
Email: louis.labuschagne@nwu.ac.za

DOI: https://doi.org/10.1090/tran/7275
Keywords: Subdiagonal operator algebra, peak projection, noncommutative Lebesgue decomposition, noncommutative Hardy space, sigma-finite von Neumann algebra, Kaplansky density theorem, F \& M Riesz theorem
Received by editor(s): January 16, 2017
Received by editor(s) in revised form: May 6, 2017
Published electronically: August 21, 2018
Additional Notes: This work is based on research supported by the National Research Foundation (IPRR Grant 96128), and the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author, and therefore the NRF does not accept any liability in regard thereto.
Article copyright: © Copyright 2018 American Mathematical Society

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