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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Brown measure and iterates of the Aluthge transform for some operators arising from measurable actions


Authors: Ken Dykema and Hanne Schultz
Journal: Trans. Amer. Math. Soc. 361 (2009), 6583-6593
MSC (2000): Primary 47A05; Secondary 47B99
Published electronically: July 20, 2009
MathSciNet review: 2538606
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Abstract: We consider the Aluthge transform $ \widetilde{T}=\vert T\vert^{1/2}U\vert T\vert^{1/2}$ of a Hilbert space operator $ T$, where $ T=U\vert T\vert$ is the polar decomposition of $ T$. We prove that the map $ T\mapsto\widetilde{T}$ is continuous with respect to the norm topology and with respect to the $ *$-SOT topology on bounded sets. We consider the special case in a tracial von Neumann algebra when $ U$ implements an automorphism of the von Neumann algebra generated by the positive part $ \vert T\vert$ of $ T$, and we prove that the iterated Aluthge transform converges to a normal operator whose Brown measure agrees with that of $ T$ (and we compute this Brown measure). This proof relies on a theorem that is an analogue of von Neumann's mean ergodic theorem, but for sums weighted by binomial coefficients.


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Additional Information

Ken Dykema
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: kdykema@math.tamu.edu

Hanne Schultz
Affiliation: Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark
Email: schultz@imada.sdu.dk

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04762-X
PII: S 0002-9947(09)04762-X
Keywords: Aluthge transform, Brown measure, mean ergodic theorem
Received by editor(s): December 12, 2005
Received by editor(s) in revised form: February 2, 2008
Published electronically: July 20, 2009
Additional Notes: An earlier version of this paper was distributed under the title: “On the Aluthge transform: continuity properties and Brown measure”.
The first author’s research was supported in part by NSF grant DMS–0300336.
As a student of the Ph.D. school OP-ALG-TOP-GEO, the second author was partially supported by the Danish Research Training Council and The Danish National Research Foundation.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.