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A unital C*-algebra A is said to have the (APD)-property if every nonzero element in A has the approximate polar decomposition. Let J be a closed ideal of A. Suppose that J and A/J have (APD). In this paper, we give a necessary and sufficient condition that makes A have (APD). Furthermore, we show that if RR(J) = 0 and tsr(A/J) = 1 or A/J is a simple purely infinite C*-algebra, then A has (APD).

(APD)–property of $C^*$–algebras by extensions of $C^*$–algebras with (APD)

(APD)–property of $C^$–algebras by extensions of $C^$–algebras with (APD)