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

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