Noncommutative maximal ergodic inequalities associated with doubling conditions

We study noncommutative maximal inequalities and ergodic theorems for group actions on von Neumann algebras. Consider a locally compact G of polynomial growth with symmetric subset V. Let ? be continuous action algebra M by trace-preserving automorphisms. then show that the operators defined Anx=1m(Vn)?Vn?gxdm(g),x?Lp(M),n?N,1?p??, are weak type (1,1) strong (p,p) 1<p<?. Consequently, sequence (Anx)n?1 converges almost uniformly x?Lp(M) 1?p<?. Also, we establish individual associated more general doubling conditions, prove corresponding results one fixed Lp-space which beyond class Dunford–Schwartz considered previously Junge Xu. As key ingredients, also obtain Hardy–Littlewood inequality metric spaces measures in operator-valued setting. After groundbreaking work Xu inequalities, this is first time proved Xu’s Our approach based quantum probabilistic methods as well random-walk theory.