Extend Rota’s theorem to noncommutative LlogL spaces

Establish the noncommutative version of Rota’s “Alternierende Verfahren” theorem by proving almost uniform convergence of the sequence T^n (T*)^n x for every x in the noncommutative Orlicz space LlogL(M, τ), where T: (M, τ) → (M, τ) is a factorizable τ-preserving Markov operator, thereby extending the L^p (p > 1) noncommutative results to the LlogL setting analogous to the classical case.

Background

Rota’s classical theorem establishes almost-everywhere convergence for products involving a Markov operator and its adjoint; in noncommutative settings, Anantharaman-Delaroche proved an analogue for L^p spaces with p > 1 using factorizable Markov operators.

The extension of this noncommutative Rota theorem from L^p (p > 1) to the endpoint Orlicz space LlogL was highlighted as an open problem in the literature, aiming to match the classical LlogL case. This paper reports that it fully establishes the theorem for noncommutative LlogL and even more general Orlicz spaces.