On the Aluthge Transform: Continuity Properties and Brown Measure
نویسندگان
چکیده
We consider the Alugthe transform T̃ = |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 7→ T̃ is continuous with respect to the norm topology and with respect to the ∗–SOT topology. For T in a tracial von Neumann algebra, we show that the Brown measure is unchanged by the Aluthge transform. We consider the special case 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 (which we prove) that is an analogue of von Neumann’s mean ergodic theorem, but for sums weighted by binomial coefficients.
منابع مشابه
Brown Measure and Iterates of the Aluthge Transform for Some Operators Arising from Measurable Actions∗
We consider the Aluthge transform T̃ = |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 7→ T̃ 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 gen...
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