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

We consider the Aluthge transform T = |T|1/2U|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. 2009 American Mathematical Society.

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