On weighted monotonicity inequalities and characterizations of the traces

Abstract

Let τ be a faithful normal semi-finite trace on a von Neumann algebra M, \(\tilde \tau \) is the extension of τ on the set of all (in general, unbounded) self-adjoint positive operators affiliated with M. We study inequalities of the form $$\tilde \tau (w(A)^{1/2} f(A)w(A)^{1/2} ) \leqslant \tilde \tau (w(A)^{1/2} f(B)w(A)^{1/2} ),$$ where f, w are nonnegative real-valued functions, A and B are unbounded self-adjoint positive operators affiliated with algebra M such that A ≤ B. A new characterization of the traces related to these inequalities will be shown.