TRACE JENSEN INEQUALITY AND RELATED WEAK MAJORIZATION IN SEMI-FINITE VON NEUMANN ALGEBRAS

摘要

Let М be a semi-finite von Neumann algebra equipped with a faithful semi-finite normal trace т, and we assume that f(t) is a convex function with f(0) = 0. The trace Jensen inequality τ (f(a*xa))≤τ(a*f(x)a)proved for a contraction a ∈ М and a self adjoint operator x ∈ М (or more generally for a semi-bounded т-measurable operator) together with an abundance of related weak majorization-type inequalities. Notions of generalized singular numbers and spectral scales are used to express our results. Monotonicity properties for the map: x ∈ M_(sa) →τ(f(x))are also invxestigated for an increasing function f (t) with f (0) = 0.