A projective C∗-algebra related to K-theory

The C∗-algebra qC is the smallest of the C∗-algebras qA introduced by Cuntz [J. Cuntz, A new look at KK-theory, K-Theory 1 (1) (1987) 31–51] in the context of KK-theory. An important property of qC is the natural isomorphism K0(A)∼= lim −→ qC,Mn(A) . Our main result concerns the exponential (boundary) map from K0 of a quotient B to K1 of an ideal I. We show if a K0 element is realized in hom(qC,B) then its boundary is realized as a unitary in I˜. The picture we obtain of the exponential map is based on a projective C∗-algebra P that is universal for a set relations slightly weaker than the relations that define qC. A new, shorter proof of the semiprojectivity of qC is described. Smoothing questions related the relations for qC are addressed. © 2008 Elsevier Inc. All rights reserved