Noncommutative multi-parameter Wiener–Wintner type ergodic theorem

Abstract

In this paper, we establish a multi-parameter version of Bellow and Losert's Wiener–Wintner type ergodic theorem for dynamical systems not necessarily commutative. More precisely, we introduce a weight class D, which is shown to strictly include the multi-parameter bounded Besicovitch weight class, thus including the setΛd={{λ1k1⋯λdkd}(k1,…,kd)∈Nd:(λ1,…,λd)∈Td}; then we prove a multi-parameter Bellow and Losert's Wiener–Wintner type ergodic theorem for the class D and for a noncommutative trace preserving dynamical system (M,τ,T), M being a von Neumann algebra. Restricted to Λd, we also prove a noncommutative multi-parameter analogue of Bourgain's uniform Wiener–Wintner ergodic theorem.The “noncommutativity” and the “multi-parameter” characters induce some difficulties in the proofs. For instance, our argument of proving the uniform convergence for a dense subset turns out to be quite different from the classical case since the “pointwise” argument does not work in the noncommutative setting; also to obtain the uniform convergence in the largest spaces, we need a maximal inequality between the Orlicz spaces, but it cannot be deduced by using classical extrapolation argument directly. Junge and Xu's noncommutative maximal inequalities with the optimal order, together with the atomic decomposition of Orlicz spaces, play the essential role in overcoming the second difficulty.