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 K-0 (A) congruent to lim(-->)[qC, M-n (A)]. Our main result concerns the exponential (boundary) map from K0 of a quotient B to K-1 of an ideal I. We show if a K-0 element is realized in hom(qC, B) then its boundary is realized as a unitary in (I) over tilde. 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 serniprojectivity of qC is described. Smoothing questions related the relations for qC are addressed. (C) 2008 Elsevier Inc. All rights reserved.
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