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

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.