Noncommutative Ergodic Theorems for Connected Amenable Groups

2016/05/11 by Sun, Mu
#Dynamical Systems (math.DS) #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1605.03568

Abstract

This paper is devoted to the study of noncommutative ergodic theorems for connected amenable locally compact groups. For a dynamical system (M,τ,G,σ), where (M,τ) is a von Neumann algebra with a normal faithful finite trace and (G,σ) is a connected amenable locally compact group with a well defined representation on M, we try to find the largest noncommutative function spaces constructed from M on which the individual ergodic theorems hold. By using the Emerson-Greenleaf's structure theorem, we transfer the key question to proving the ergodic theorems for ℝ^d group actions. Splitting the ℝ^d actions problem in two cases according to different multi-parameter convergence types---cube convergence and unrestricted convergence, we can give maximal ergodic inequalities on L₁(M) and on noncommutative Orlicz space L₁log^2(d-1)L(M), each of which is deduced from the result already known in discrete case. Finally we give the individual ergodic theorems for G acting on L₁(M) and on L₁log^2(d-1)L(M), where the ergodic averages are taken along certain sequences of measurable subsets of G.

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