A trace interpolation and characterizations of tracial positive linear functionals

Let τ be a faithful normal semifinite trace on a von Neumann algebra M. For A,B∈L1(M,τ)+, the quantity τ(|A−VBW|), where V,W∈M are contractions, interpolates between the two extremes τ(A−B) and τ(A+B). Inspired by this interpolation, we prove that a positive linear functional φ on M is tracial if and only if φ(A−B)≤φ(|A−VBW|) for all A,B∈M+ and all contractions V and W in M. We also strengthen L. T. Gardner's trace characterization in the context of von Neumann algebras. Motivated by a different trace inequality, we establish that a normal state φ on a von Neumann algebra M is tracial if and only if it satisfies φ(ABA)≤φ((A+B2)2) for all projections A,B∈M. Furthermore, we introduce a new criterion for identifying commutative von Neumann algebras based on this inequality.