Extensions of C ( X ) by Simple C *-algebras of real rank zero

Abstract

Let Ext ( C ( X ), A ) be the set of unitarily equivalence classes of essential C *-algebra extensions of the following form: 0 → A → E → C ( X ) → 0, where A is a nonunital separable simple C *-algebra of real rank zero, stable rank one with unique normalized trace and X is a finite CW complex. We show that there is a bijection [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]: Ext ( C ( X ), A ) → KK ( C ( X ), M ( A )/ A ), where M ( A ) is the multiplier algebra of A . In particular, we determine when an extension is actually splitting. We also, in a more general setting, give a condition when an essential extension is quasidiagonal.

Simple corona $C^*$-algebras

“…In this short note we first show that in fact the converse also holds: if A ∼ = K is a non-unital and σ-unital simple C * -algebra such that M (A)/A is simple, then A has a continuous scale. The renewed interests in the corona algebras M (A)/A are related to the classification of nuclear C * -algebras and the study of essential extensions by simple C * -algebras (see for example, [Ln2] and [Ln4]). …”

Quasidiagonal Morphisms and Homotopy

“…Thus one may argue that quasidiagonality is related to K-theoretical phenomena. For more on the connections of quasidiagonal to K-theory see [Br,Sa 1,2 ,Zh,BrD,Li,Sc].…”

Full extensions and approximate unitary equivalence

“…When B is a non-unital but σ-unital simple C * -algebra with continuous scale (see (6) below), then M (B)/B is simple. Classification of essential extensions of a separable amenable C *algebra A by B (up to approximate unitary equivalence) was obtained in [32] (for some special cases in which A = C(X), classification up to unitary equivalence was obtained in [22], [23] and [25]). In this case, B may not be stable, therefore KK 1 (A, B) is not used as invariant for essential extensions.…”