Noncommutative maximal ergodic theorems

Abstract

This paper is devoted to the study of various maximal ergodic theorems in noncommutative L p L_p -spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on L p L_p and the analogue of Stein’s maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in von Neumann algebra theory and in quantum probability.

Noncommutative strong maximals and almost uniform convergence in several directions

“…The formulation of noncommutative maximal inequalities is already subtle, since it is not possible to define the supremum of a family of noncommuting operators in a meaningful way. Namely, as in the Introduction of [28], one can produce examples of 2 × 2 noncommuting matrices 1 , 2 , 3 for which no 2 × 2 matrix satisfies , = max{ , : = 1, 2, 3} for all ∈ R 2 . The trick to overcome this for weak type maximal inequalities is to use…”

“…It was Junge who extended in 2002 Doob's -maximal inequality for noncommutative martingales with an ad hoc argument heavily relying on Hilbert module theory and duality [24]. A few years later, Junge and Xu obtained -maximal inequalities for ergodic means and subordinated Markovian semigroups [28]. The key point was a novel interpolation theorem for families of positive preserving maps-as a substitute for Marcinkiewicz interpolation-which allows one to infer results from Cuculescu and Yeadon's 'extra-weak' inequalities.…”

“…Later, this notion was extended to the setting of QWEP von Neumann algebras by Junge [25]. In the case of = ℓ ∞ , an ad hoc definition requiring a factorization was introduced in [24] (see also [28]). In this case, one does not need any approximation properties of the von Neumann algebra.…”

“…(see [28]). Another property that will be useful is the fact that (M; ℓ ∞ ) is a bimodule with respect to M: that is,…”

“…Two tools that we are going to use repeatedly are the noncommutative analogues of Marcinkiewicz interpolation and Yano's extrapolation [50] in the case of maximal operators. The particular noncommutative analogue of Marcinkiewicz interpolation that we will use was obtained in [28] (see also [8]), while the noncommutative extrapolation was obtained in [20]. Let us start by recalling the following definition, which extends the classical notion of weak type ( , ) to von Neumann algebras.…”