Voiculescu theorem, Sobolev lemma, and extensions of smooth algebras

Wang, Xiaolu

doi:10.1090/S0273-0979-1992-00326-9

Voiculescu theorem, Sobolev lemma, and extensions of smooth algebras
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Bull. Amer. Math. Soc. 27 (1992), 292-297 Request permission

Abstract:

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 Fréchet operator ideal ${\mathcal {K}_\tau }$. 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 $\tau$-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.

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