Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4592783 | Journal of Functional Analysis | 2008 | 14 Pages |
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.