Ergodic theorems in fully symmetric spaces of τ-measurable operators

Abstract

In [11], employing the technique of noncommutative interpolation, a maximal ergodic theorem in noncommutative Lp−spaces, 1 < p < ∞, was established and, among other things, corresponding maximal ergodic inequalities and individual ergodic theorems were derived. In this article, we derive maximal ergodic inequalities in noncommutative Lp−spaces directly from [25] and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these results to noncommutative fully symmetric Banach spaces with Fatou property and non-trivial Boyd indices, in particular, to noncommutative Lorentz spaces Lp,q. Norm convergence of ergodic averages in noncommutative fully symmetric Banach spaces is also studied.Date: February 8, 2015. 2010 Mathematics Subject Classification. 47A35(primary), 46L52(secondary).

Almost uniform convergence in the noncommutative Dunford–Schwartz ergodic theorem

“…(respectively, a.u.). Later, in [10] (see also [2]), it was shown that this result can be obtained directly from Yeadon's maximal inequality for L 1 (M, τ ) established in [21]. In particular, it was shown that a.u.…”

“…Given 1 ≤ p < ∞, let a sequence {y m } ⊂ L p be such that y m p → 0. Then for any ǫ > 0 and δ > 0 there exist a projection e ∈ P(M) and a sequence {x k } with x k ∈ {y m } for every k such that (2) τ (e ⊥ ) ≤ ǫ and sup n A n (x k )e ∞ ≤ δ for all k.…”

“…But the problem persisted, and a significant number of noncommutative individual ergodic theorems concerning b.a.u. convergence of ergodic averages have been established; see, for example, [15,5,3,11,7,18,12,2].…”

“…But the problem persisted, and a significant number of noncommutative individual ergodic theorems concerning b.a.u. convergence of ergodic averages have been established; see, for example, [15,5,3,11,7,18,12,2].It was derived in [7] that if 1 < p < 2 (2 ≤ p < ∞), then for a positive DunfordSchwartz operator in a noncommutative space L p (M, τ ), the corresponding ergodic Cesáro averages converge b.a.u. (respectively, a.u.).…”

Ergodic theorems in Banach ideals of compact operators

“…The study of individual ergodic theorems beyond L 1 (M, τ ) started much later with another fundamental paper by M. Junge and Q. Xu [13], where, among other results, individual ergodic theorem was extended to the case with a positive Dunford-Schwartz operator acting in the space L p (M, τ ), 1 < p < ∞. In [3] ( [4]), utilizing the approach of [16], an individual ergodic theorem was proved for a positive Dunford-Schwartz operator in a noncommutative Lorentz (respectively, Orlicz) space.…”