$S$-regularity and the corona factorization property

Abstract

Stability is an important and fundamental property of $C^{*}$-algebras. Given a short exact sequence of $C^{*}$-algebras $0\longrightarrow B\longrightarrow E\longrightarrow A\longrightarrow 0$ where the ends are stable, the middle algebra may or may not be stable. We say that the first algebra, $B$, is $S$-regular if every extension of $B$ by a stable algebra $A$ has a stable extension algebra, $E$. Rördam has given a sufficient condition for $S$-regularity. We define a new condition, weaker than Rördam's, which we call the corona factorization property, and we show that the corona factorization property implies $S$-regularity. The corona factorization property originated in a study of the Kasparov $KK^1(A,B)$ group of extensions, however, we obtain our results without explicit reference to $KK$-theory. Our main result is that for a separable stable $C^{*}$-algebra $B$ the first two of the following properties (which we define later) are equivalent, and both imply the third. With additional hypotheses on the $C^{*}$-algebra, all three properties are equivalent. $B$ has the corona factorization property. Stability is a stable property for full hereditary subalgebras of $B$. $B$ is $S$-regular. We also show that extensions of separable stable $C^{*}$-algebras with the corona factorization property give extension algebras with the corona factorization property, extending the results of [9].