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## BROWN MEASURE AND ITERATES OF THE ALUTHGE TRANSFORM FOR SOME OPERATORS ARISING FROM MEASURABLE ACTIONS ∗

### Abstract

Abstract. We consider the Aluthge transform � 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 ↦ → � 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. 1

Year: 2009
OAI identifier: oai:CiteSeerX.psu:10.1.1.134.7742
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