Brown measure and iterates of the Aluthge transform for some operators arising from measurable actions
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- by Ken Dykema and Hanne Schultz PDF
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Abstract:
We consider the Aluthge transform $\widetilde {T}=|T|^{1/2}U|T|^{1/2}$ of a Hilbert space operator $T$, where $T=U|T|$ 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 $|T|$ 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.References
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Additional Information
- Ken Dykema
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- MR Author ID: 332369
- 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
- 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. - © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 6583-6593
- MSC (2000): Primary 47A05; Secondary 47B99
- DOI: https://doi.org/10.1090/S0002-9947-09-04762-X
- MathSciNet review: 2538606