Failure of noncommutative Rota's theorem with $L\log L$ endpoint

Abstract

We construct a factorizable Markov map $T$ on a hyperfinite finite von Neumann algebra and a positive element $x\in L\log L$ for which the sequence $T^n(T^{\ast }{)}^{n}x$ does not converge bilaterally almost uniformly. This answers a question of Anantharaman-Delaroche \cite{AD2006}.

arXiv comment16 pages