标题： Voiculescu定理，Sobolev引理，以及光滑代数的扩展 显示英文标题

标题： Voiculescu theorem, Sobolev lemma, and extensions of smooth algebras

摘要： 我们提出了非交换光滑代数范畴上统一的B-D-F扩展函子$\operatorname{Ext}_\tau$的解析基础，对于任何Fréchet算子理想$\Cal K_\tau$。 结合Arveson和Voiculescu设计的技术，我们将Voiculescu定理推广到光滑代数和Fréchet算子理想。 涉及的一个关键概念是$\tau$-光滑性，通过非交换Sobolev引理验证了光滑函数代数的这一性质。 许多例子中的群$\operatorname{Ext}_\tau$都被计算出来了。 显示英文摘要

摘要： 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.