Projections and Traces on von Neumann Algebras

Abstract

© 2019, Pleiades Publishing, Ltd. 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 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 ∈ 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 ∈ 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 M the following conditions are equivalent: (i) φ is tracial; (ii) φ(A + A*) ≤ 2φ(∣A*∣) for all A ∈ M.