We continue the
investigation of the extension theory of smooth algebras in the framework of
quantized differential geometry. Here we compute more examples of smooth
extensions starting with the cases of dimensions 0 and 1. In particular, we prove a
vanishing theorem for totally disconnected spaces. We show that for any compact
smooth manifold of positive dimension, there is a representation of C(M) defining
a degenerate ℒ1-smooth extension which is not ℒ1-smooth. However for
any Fréchet operator ideal 𝒦τ, we prove that all completely positive maps
defining extensions of C∞(S1) by 𝒦τ are τ-smooth, and that all the groups
Extτ(C∞(S1)) ≃ ℤ.
