Almost uniform convergence in the noncommutative Dunford–Schwartz ergodic theorem

This article gives an affirmative solution to the problem whether the ergodic Cesáro averages generated by a positive Dunford–Schwartz operator in a noncommutative space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="script">M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mo>∞</mml:mo> </mml:math> , converge almost uniformly (in Egorov's sense). This problem goes back to the original paper of Yeadon [21], published in 1977, where bilaterally almost uniform convergence of these averages was established for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> .

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