Notes on noncommutative ergodic theorems

2023/08/03 by Litvinov, Semyon
#46L51(secondary) #47A35(primary) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2308.02578

Abstract

Given a semifinite von Neumann algebra \mathcal M equipped with a faithful normal semifinite trace τ, we prove that the spaces L⁰(\mathcal M,τ) and \mathcal R_τ are complete with respect to pointwise, almost uniform and bilaterally almost uniform, convergences in L⁰(\mathcal M,τ). Then we show that the pointwise Cauchy property for a special class of nets of linear operators in the space L¹(\mathcal M,τ) can be extended to pointwise convergence of such nets in any fully symmetric space E⊂\mathcal R_τ, in particular, in any space L^p(\mathcal M,τ), 1≤ p<∞. Some applications of these results in the noncommutative ergodic theory are discussed.

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