Metrics on projections of the von neumann algebra associated with tracial functionals

Let φ be a positive functional on a von Neumann algebra \(\mathscr{A}\) and let \(\mathscr{A}^{pr}\) be the projection lattice in \(\mathscr{A}\). Given \(P,Q \in \mathscr{A}^{pr}\), 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 \(\mathscr{A}^{pr}\). Moreover, if τ is normal then (\(\mathscr{A}^{pr}\), ρ$_{τ}$) and (\(\mathscr{A}^{pr}\), 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.