On Uniform Equicontinuity of Sequences of Measurable Operators

TL;DR

This paper investigates uniform equicontinuity in measure for sequences of measurable operators in a non-commutative setting, establishing conditions under which local properties extend globally and applying these results to ergodic theorems.

Contribution

It introduces conditions for extending uniform equicontinuity from dense subsets to entire spaces in the context of measurable operators.

Findings

Uniform equicontinuity in measure at zero on dense subsets implies global uniform equicontinuity.

Application of these results to derive non-commutative ergodic theorems.

Abstract

The notion of uniform equicontinuity in measure at zero for sequences of additive maps from a normed space into the space of measurable operators associated with a semifinite von Neumann algebra is discussed. It is shown that uniform equicontinuity in measure at zero on a dense subset implies the uniform equicontinuity on the entire space, which is then applied to derive some non-commutative ergodic theorems