TY - JOUR

T1 - A noncommutative generalisation of a problem of Steinhaus

AU - Junge, Marius

AU - Scheckter, Thomas Tzvi

AU - Sukochev, Fedor

N1 - Funding Information:
The second named author was supported by an Australian Government Research Training Program (RTP) Scholarship.The third named author was supported by ARC Laureate Fellowship grant FL170100052 — Breakthrough methods for noncommutative calculus.
Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2021/1/15

Y1 - 2021/1/15

N2 - We extend the Révész and Komlós theorems to arbitrary finite von Neumann algebras, and in doing so solve an open problem of Randrianantoanina, removing the need for hyperfiniteness. The main result is the noncommutative Komlós theorem, which states that every norm-bounded sequence of operators in L1(M), for any finite von Neumann algebra M, admits a subsequence, such that for any further subsequence, the Cesàro averages converge bilaterally almost uniformly. This is a natural extension of Komlós' original result to the noncommutative setting. The necessary techniques which facilitate the proof also allow us to extend the Révész theorem to the noncommutative setting, which gives a similar subsequential law for series over bounded sequences in L2(M).

AB - We extend the Révész and Komlós theorems to arbitrary finite von Neumann algebras, and in doing so solve an open problem of Randrianantoanina, removing the need for hyperfiniteness. The main result is the noncommutative Komlós theorem, which states that every norm-bounded sequence of operators in L1(M), for any finite von Neumann algebra M, admits a subsequence, such that for any further subsequence, the Cesàro averages converge bilaterally almost uniformly. This is a natural extension of Komlós' original result to the noncommutative setting. The necessary techniques which facilitate the proof also allow us to extend the Révész theorem to the noncommutative setting, which gives a similar subsequential law for series over bounded sequences in L2(M).

KW - Komlós theorem

KW - Noncommutative integration

KW - Strong law of large numbers

KW - Subsequence principle

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U2 - 10.1016/j.jfa.2020.108782

DO - 10.1016/j.jfa.2020.108782

M3 - Article

AN - SCOPUS:85092118738

SN - 0022-1236

VL - 280

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 2

M1 - 108782

ER -