S-regularity and the corona factorization property - Kucerovsky, D.; Ng, P. W. (MATHEMATICA SCANDINAVICA, 2006)
Stability is an important and fundamental property of C^*-algebras. Given a short exact sequence of C^*-algebras 0BEA0 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].