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We show that the following properties of the \({\rm C^*}\)-algebras in a class \(\Omega\) are inherited by simple unital \({\rm C^*}\)-algebras in the class \({\rm TA}\Omega\): \((1)\) \(\beta\)-comparison (\(1\leq \beta < \infty\)), \((2)\) \(n\)-comparison, \((3)\) trace \(\mathcal{Z}\)– absorption, \((4)\) \(m\)-almost divisibility, \((5)\) \((n,m) ~(m\neq 0)\) comparison, and \((6)\) tracial approximate divisibility. As an application, every unital simple \({\rm C^*}\)-algebra with tracial topological rank at most \(k\) has the property of \(k\)-comparison. Also as an application, let \(A\) be an infinite-dimensional simple unital \({\rm C^*}\)-algebra such that \(A\) has one of the above-listed properties. Suppose that \(\alpha: G\to {\rm Aut}(A)\) is an action of a finite group \(G\) on \(A\) which has the tracial Rokhlin property. Then the crossed product \({\rm C^*}\)-algebra \({\rm C^*}( G, A,\alpha)\) also has the property under consideration.

On considère plusieurs propriétés d’une C*-algèbre simple à élément unité qui sont héritées par approximation traciale. Comme application on démontre que ces propriétés sont aussi héritées par la C*-algèbre produit croisé associée à une action d’un groupe fini qui possède la propriété de Rokhlin traciale.