Trace inequalities for Rickart C^∗ -algebras

Abstract

Rickart C∗-algebras are unital and satisfy polar decomposition. We proved that if a unital C∗-algebra A satisfies polar decomposition and admits “good” faithful tracial states then A is a Rickart C∗-algebra. Via polar decomposition we characterized tracial states among all states on a Rickart C∗-algebra. We presented the triangle inequality for Hermitian elements and traces on Rickart C∗-algebra. For a block projection operator and a trace on a Rickart C∗-algebra we proved a new inequality. As a corollary, we obtain a sharp estimate for a trace of the commutator of any Hermitian element and a projection. Also we give a characterization of traces in a wide class of weights on a von Neumann algebra.