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

Abstract

Let $0\longrightarrow \B\stackrel{j}{\longrightarrow}E\stackrelπ{\longrightarrow}\A\longrightarrow 0$ be an extension of $\A$ by $\B$, where $\A$ is a unital simple purely infinite $C^{\ast }$--algebra. When $\B$ is a simple separable essential ideal of the unital $C^{\ast }$--algebra $E$ with $\RR(\B)=0$ and {\rm(PC)}, $K_0(E)=\{[p]\mid p$ is a projection in $E\setminus B\}$; When $B$ is a stable $C^{\ast }$--algebra, $\U(C(X,E))/\U_0(C(X,E))\cong K_1(C(X,E))$ for any compact Hausdorff space $X$.

7 pages, Chinese Math. Annal (Chinese) (accepted)