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EB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 $X$.