Projections and Traces on von Neumann Algebras

Abstract

Let P, Q be projections on a Hilbert space. We prove the equivalence of the following conditions: (i) PQ + QP ≤ 2(QPQ)^p for some number 0 < p ≤ 1; (ii) PQ is paranormal; (iii) PQ is M*-paranormal; (iv) PQ = QP. This allows us to obtain the commutativity criterion for a von Neumann algebra. For a positive normal functional φ on von Neumann algebra \(\mathcal{M}\) it is proved the equivalence of the following conditions: (i) φ is tracial; (ii) φ(PQ + QP) ≤ 2φ((QPQ)^p) for all projections P,Q ∈ \(\mathcal{M}\) and for some p = p(P, Q) ∈ (0,1]; (iii) φ(PQP) ≤ φ(P)^1/pφ(Q)^1/q for all projections P, Q ∈ \(\mathcal{M}\) and some positive numbers p = p(P, Q), q = q(P, Q) with 1/p+ 1/q = 1, p ≠ 2. Corollary: for a positive normal functional φ on \(\mathcal{M}\) the following conditions are equivalent: (i) φ is tracial; (ii) φ(A + A*) ≤ 2φ(∣A*∣) for all A ∈ \(\mathcal{M}\).