Probability measures in W* J-algebras in Hilbert spaces with conjugation

Abstract

Let M be a real W*-algebra of J-real bounded operators containing no central summand of type I2 in a complex Hubert space H with conjugation J. Denote by P the quantum logic of all J-orthogonal projections in the von Neumann algebra N = M + iM. Let μ : P → [0,1] be a probability measure. It is shown that H contains a finite central summand and there exists a normal finite trace τ on N such that μ(p) = τ(p), p ∈ P. © 1998 American Mathematical Society.