Metrics on Projections of the Von Neumann Algebra Associated with Tracial Functionals

Abstract

Let φ be a positive functional on a von Neumann algebra \(\mathscr{A}\) and let \(\mathscr{A}^{\rm{pr}}\) be the projection lattice in \(\mathscr{A}\). Given \(P,Q \in \mathscr{A}^{\rm{pr}}\), put ρ_φ(P, Q) = φ(∣P − Q∣) and d_φ(P, Q) = φ(P ∨ Q − P ∧ Q). Then ρ_φ(P, Q) ≤ d_φ(P, Q) and ρ_φ(P, Q) = d_φ(P, Q) provided that PQ = QP. The mapping ρ_φ (or d_φ) meets the triangle inequality if and only if φ is a tracial functional. If τ is a faithful tracial functional then ρ_τ and d_τ are metrics on \(\mathscr{A}^{\rm{pr}}\). Moreover, if τ is normal then (\(\mathscr{A}^{\rm{pr}}\), ρ_τ) and (\(\mathscr{A}^{\rm{pr}}\), d_τ) are complete metric spaces. Convergences with respect to ρ_τ and d_τ are equivalent if and only if \(\mathscr{A}\) is abelian; in this case ρ_τ = d_τ. We give one more criterion for commutativity of \(\mathscr{A}\) in terms of inequalities.