Department of Mathematics,

Pennsylvania State University,

Hazleton, PA 18202,

USA

and

National University of Uzbekistan,

Tashkent 700095,

UZBEKISTAN

E-mail: snl2@psu.edu, chilin@ucd.uz

**A Banach Principle for a von Neumann Algebra**

**Abstract:**

For a sequence of homomorphisms from a finite von Neumann algebra *M*
to the algebra of all measurable operators affiliated with *M*, we
introduce a notion of uniform equicontinuity (with respect to the measure
topology). Then we discuss its relationship with the almost uniform convergence
of a sequence of continuous homomorphisms.