Characterization of Tracial Functionals on Von Neumann Algebras

Abstract

Abstract: It is proved that the inequality (Formula presented.) characterizes tracial functionals among all positive normal functionals ϕ on a von Neumann algebra A. This strengthens the L. T. Gardner’s characterization (1979). As a consequence, a criterion for commutativity of von Neumann algebras is obtained. Also we give a characterization of traces in a wide class of weights on a von Neumann algebra via this inequality. Every faithful normal semifinite trace ϕ on a von Neumann algebra A satisfies this relation. Let (Formula presnted.) be a unitarily invariant norm on a unital C*-algebra A. Then (Formula presnted.) for all A ϵ A+ and B ϵ Asa.