  We consider the Aluthge transform $|T|^{1/2}U|T|^{1/2}$ of a Hilbert space
operator $T$, where $T=U|T|$ is the polar decomposition of $T$. We prove that
the map that sends $T$ to its Aluthge transform 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.
