Noncommutative phenomena in measure theory on operators algebras

Abstract

The aim of this paper is to to review the basic principles of the measure theory built on operator algebras and to summarize recent progress in this field. We concentrate on those results which demonstrate considerable difference between the classical measure theory and its operator-algebraic counterpart. In particular, we show that the functional-analytic structure of a a given operator algebra A, the lattice-theoretic properties of measures on the projection lattice of A, the facial structure of the convex set of all states on A, and the continuity properties of states on A are connected in a way has no analogy in the standard measure theory.