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

Abstract

Let script M be a real semifinite W*-algebra of J-real operators containing no finite central summand in a complex Hilbert space H with conjugation J. Denote by P the quantum logic of all J-orthogonal projections in the von Neumann algebra script N = script M + iscript N. Let μ: P → R be a Hermitian measure. It is shown that there exists an unique J-self-adjoint ultraweakly continuous linear functional ψ on script N such that μ(p) =ℛψ(p), ∀p ∈ P.