Semyon Litvinov

Uniform equicontinuity of sequences of measurable operators and non-commutative ergodic theorems

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 in measure at zero on the entire space, which is then applied to derive some non-commutative ergodic theorems.

Individual ergodic theorems in noncommutative Orlicz spaces

For a noncommutative Orlicz space associated with a semifinite von Neumann algebra, a faithful normal semifinite trace and an Orlicz function satisfying

Almost uniform and strong convergences in ergodic theorems for symmetric spaces

(\delta _2,\Delta _2)

Ergodic theorems in fully symmetric spaces of τ-measurable operators

(δ2,Δ2)-condition, an individual ergodic theorem is proved.

The validity space of Dunford–Schwartz pointwise ergodic theorem

Utilizing the notion of uniform equicontinuity for sequences of functions with the values in the space of measurable operators, we present a non-commutative version of the Banach Principle for L 1 .

Almost uniform convergence in the noncommutative Dunford–Schwartz ergodic theorem

Let

Pointwise ergodic theorems in symmetric spaces of measurable functions

{(\Omega,\mu)}