Trace inequalities for Rickart $$C^*$$ C ∗ -algebras

Rickart $$C^*$$ -algebras are unital and satisfy polar decomposition. We proved that if a unital $$C^*$$ -algebra $${\mathcal {A}}$$ satisfies polar decomposition and admits “good” faithful tracial states then $${\mathcal {A}}$$ 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.