Hermitian Measures in W*J-Algebras in Hilbert Spaces with Conjugation

Abstract

Let M be a real semifinite W ^*-algebra of J-real operators containing no finitecentral summand in a complex Hilbert space H with conjugation J. Denote byP the quantum logic of all J-orthogonal projections in the von Neumann algebraN = M + i>M. Let μ: P → R be a Hermitian measure. It is shown that thereexists an unique J-self-adjoint ultraweakly continuous linear functional ψ on Nsuch that μ(p) = Rψ(p), ??p ∈ P.