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

TL;DR

This paper investigates the non-stable K-theory of extension algebras formed by a simple purely infinite C*-algebra and certain other C*-algebras, providing new insights into their structure and unitary groups.

Contribution

It introduces new results on the K-theory of extension algebras involving simple purely infinite C*-algebras and characterizes their unitary groups under specific conditions.

Findings

K_0(E) characterized by projections outside B

U(C(X,E))/U_0(C(X,E)) is isomorphic to K_1(C(X,E)) for stable B

Extension algebras exhibit specific K-theoretic properties

Abstract

Let $0\longrightarrow \B\stackrel{j}{\longrightarrow}E\stackrel{\pi}{\longrightarrow}\A\longrightarrow 0$ be an extension of $\A$ by $\B$, where $\A$ is a unital simple purely infinite $C^{*}$--algebra. When $\B$ is a simple separable essential ideal of the unital $C^{*}$--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 $C^{*}$--algebra, $\U(C(X,E))/\U_0(C(X,E))\cong K_1(C(X,E))$ for any compact Hausdorff space $X$.