A projective C∗C∗-algebra related to K-theory

The C∗C∗-algebra qCqC is the smallest of the C∗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 qCqC is the natural isomorphismK0(A)≅lim→[qC,Mn(A)]. Our main result concerns the exponential (boundary) map from K0K0 of a quotient B   to K1K1 of an ideal I  . We show if a K0K0 element is realized in hom(qC,B)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∗C∗-algebra PP that is universal for a set relations slightly weaker than the relations that define qCqC. A new, shorter proof of the semiprojectivity of qCqC is described. Smoothing questions related the relations for qCqC are addressed.