纯无限单的C*-代数通过某些C*-代数扩张的非稳定K-理论

设0→B■E■A→0是有单位元C~*-代数E的一个扩张,其中A是有单位元纯无限单的C~*-代数,B是E的闭理想.当B是E的本性理想并且同时是单的、可分的而且具有实秩零及性质(PC)时,证明了K_0(E)={p]| p是E\B中的投影};当B是稳定C~*-代数时,证明了对任意紧的Hausdorff空间X,有■(C(X,E))/■_0(C(X,E))≌K_1(C(X,E)).

Nonstable K-theory for Extension Algebras of the Simple Purely Infinite C*-algebra by Certain C*-algebras

Let $0 \rightarrow \B \stackrel{j}{\rightarrow} E \stackrel{\pi}{\rightarrow} \A \rightarrow 0$ be a short exact sequence of the unital $C^{*}$-algebras, where $\A$ is a unital simple purely infinite $C^{*}$-algebra, $\B$ is a closed ideal of the unital $C^{*}$-algebra $E$. If $\B$ is an essential ideal of $E$ and $\B$ is also simple, separable with $\RR(\B)=0$ and (PC), then $K_{0}(E)=\{p] \mid p$ is a projection in $E \setminus \B\}$; if $B$ is a stable $C^{*}$-algebra, then $\U(C(X,E))/\U_{0}(C(X,E)) \cong K_{1}(C(X,E))$ for any compact Hausdorff space $X$.