Projections and Traces on von Neumann Algebras

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 ) ≤ 2 φ (( QPQ ) p ) for all projections P,Q ∈ M and for some p = p ( P , Q ) ∈ (0,1]; (iii) φ ( PQP ) ≤ φ ( P ) 1/ p φ ( 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 *) ≤ 2 φ (∣ A*∣ ) for all A ∈ M .