Noncommutative Ergodic Theorems for Connected Amenable Groups

This paper is devoted to the study of noncommutative ergodic theorems for connected amenable locally compact groups. For a dynamical system $(\mathcal{M},\tau ,G,\sigma )$, where $(\mathcal{M},\tau )$ is a von Neumann algebra with a normal faithful finite trace and $(G,\sigma )$ is a connected amenable locally compact group with a well defined representation on $\mathcal{M}$, we try to find the largest noncommutative function spaces constructed from $\mathcal{M}$ 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 ${\mathbb{R}}^d$ group actions. Splitting the ${\mathbb{R}}^d$ actions problem in two cases according to different multi-parameter convergence types—cube convergence and unrestricted convergence, we can give maximal ergodic inequalities on $L_1(\mathcal{M})$ and on noncommutative Orlicz space $L_1{\log }^{2(d-1)}L(\mathcal{M})$, each of which is deduced from the result already known in discrete case. Finally we give the individual ergodic theorems for $G$ acting on $L_1(\mathcal{M})$ and on $L_1{\log }^{2(d-1)}L(\mathcal{M})$, where the ergodic averages are taken along certain sequences of measurable subsets of $G$.