Ergodic theorems in fully symmetric spaces of $\tau $-measurable operators

Vladimir Chilin ¹   ; Semyon Litvinov ²

¹ National University of Uzbekistan Tashkent, 700174, Uzbekistan
² Pennsylvania State University Hazleton, PA 18202, U.S.A.

Studia Mathematica, Tome 228 (2015) no. 2, pp. 177-195

Résumé

Junge and Xu (2007), employing the technique of noncommutative interpolation, established a maximal ergodic theorem in noncommutative $L_p$-spaces, $1 p \infty $, and derived corresponding maximal ergodic inequalities and individual ergodic theorems. In this article, we derive maximal ergodic inequalities in noncommutative $L_p$-spaces directly from the results of Yeadon (1977) and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these results to noncommutative fully symmetric Banach spaces with the Fatou property and nontrivial Boyd indices, in particular, to noncommutative Lorentz spaces $L_{p,q}$. Norm convergence of ergodic averages in noncommutative fully symmetric Banach spaces is also studied.

Détail

DOI : 10.4064/sm228-2-5

Keywords: junge employing technique noncommutative interpolation established maximal ergodic theorem noncommutative p spaces infty derived corresponding maximal ergodic inequalities individual ergodic theorems article derive maximal ergodic inequalities noncommutative p spaces directly results yeadon apply prove corresponding individual besicovitch weighted ergodic theorems extend these results noncommutative fully symmetric banach spaces fatou property nontrivial boyd indices particular noncommutative lorentz spaces norm convergence ergodic averages noncommutative fully symmetric banach spaces studied