  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.
