Nonstable $K$--theory for extension algebras of the simple purely infinite $C^{\ast }$--algebra by certain $C^{\ast }$--algebras

Let $0\longrightarrow \B\stackrel{j}{\longrightarrow}E\stackrel{\pi}{\longrightarrow}\A\longrightarrow 0$beanextensionof$\A$by$\B$,where$\A$ is a unital simple purely infinite $C^{\ast }$–algebra. When $\text{B}$ is a simple separable essential ideal of the unital $C^{\ast }$–algebra $E$ with $\text{RR}(\text{B})=0$ and {\rm(PC)}, $K_0(E)=\{[p]\mid p$ is a projection in $E\setminus B\}$; When $B$ is a stable $C^{\ast }$–algebra, $\text{U}(C(X,E))\mathrm{/}{\text{U}}_0(C(X,E))\cong K_1(C(X,E))$ for any compact Hausdorff space $X$.