Voiculescu theorem, Sobolev lemma, and extensions of smooth algebras

Abstract

We present the analytic foundation of a unified B-D-F extension functor $\operatorname{Ext}_τ$ on the category of noncommutative smooth algebras, for any Fréchet operator ideal $\Cal K_τ$. Combining the techniques devised by Arveson and Voiculescu, we generalize Voiculescu's theorem to smooth algebras and Fréchet operator ideals. A key notion involved is $τ$-smoothness, which is verified for the algebras of smooth functions, via a noncommutative Sobolev lemma. The groups $\operatorname{Ext}_τ$ are computed for many examples.