UNIFORM EQUICONTINUITY FOR SEQUENCES OF HOMOMORPHISMS INTO THE RING OF MEASURABLE OPERATORS

Abstract. We introduce a notion of uniform equicontinuity for sequences of func-tions with the values in the space of measurable operators. Then we show that all the implications of the classical Banach Principle on the almost everywhere convergence of sequences of linear operators remain valid in a non-commutative setting. Let (Ω,Σ, µ) be a probability space. Denote by L = L(Ω,Σ, µ) the set of all (classes of) complex-valued measurable functions on Ω. Let τµ stand for the measure topology in L. The classical Banach Principle may be stated as follows. Let (X, ‖ · ‖) be a Banach space, and let an: (X, ‖ · ‖) → (L, τµ) be a sequence o