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

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.