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