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\begin{document}$$C^*$$\end{document}-algebras are unital and satisfy polar decomposition. We proved that if a unital C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-algebra A\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}}$$\end{document} satisfies polar decomposition and admits “good” faithful tracial states then A\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {A}}$$\end{document} is a Rickart C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-algebra. Via polar decomposition we characterized tracial states among all states on a Rickart C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-algebra. We presented the triangle inequality for Hermitian elements and traces on Rickart C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-algebra. For a block projection operator and a trace on a Rickart C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-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.
