Extensions By C * ‐Algebras of Real Rank Zero, II

Let Ext ( C ( X ), A ) be the set of unitary equivalence classes of (unital) essential C * ‐algebra extensions of the following form: 0 → A → E → C ( X ) → 0 , where A is a simple AF‐algebra with continuous scale and X is a compact subset of the plane. We show that there is a bijection Γ : E x t ( C ( X ) , A ) → Hom 0 ( K * ( C ( X ) ) , K * ( M ( A ) / A ) ) , where Hom 0 ( K * ( C ( X )), K * ( M ( A )/ A )) is the set of homomorphisms from the graded K ‐groups of C ( X ) into the graded K ‐groups of M ( A )/ A which map the image of the unit of C ( X ) in K * of C ( X ) to the image of the unit of M ( A )/ A in K * of M ( A )/ A . We in fact show this for a larger class of C * ‐algebras. The C * ‐algebras that we consider here are not stable, nevertheless K * ( M ( A )/ A ) is computable. Some specific computation will be given. For example, we show that if X is path connected and A is a finite matroid algebra, then there is only one equivalence class in Ext ( C ( X ), A ).