Projective C-algebras and Boundary Maps

Both boundary maps in K-theory are expressed in terms of surjections from projective C∗-algebras to semiprojective C∗-algebras. 1. Noncommutative Cells and Boundaries Cells are absolute retracts that tie together spheres of different dimensions. The analog of an absolute retract for a C-algebra is being projective. For better or worse, in the category of all C-algebras, we lose the projectivity of C0 (D \ {−1}) , so we cannot generally use the exactness of 0 → C0 (R2) → C0 (D \ {−1}) → C0 (R) → 0 to explain the index map inK-theory. Another difficulty is that we need asymptotic morphisms to obtain the natural isomorphism [[ C0 (R2) , D ⊗ K]] ∼= K0(D). The name“index map”is related to the Toeplitz algebra T and the exact sequence 0 → K→ T → C(S) → 0. We might prefer to use T0, generated by the shift minus one, and 0 → K→ T0 → C0 (R) → 0, but still we may have trouble since T0 is not projective and K is not semiprojective. The “second standard picture of the index map” in [8, Proposition 9.2.2] and the picture of the exponential map presented in [6] both use what might be called noncommutative cells. In both cases, there is a diagram (1) 0 U ι P η R 0