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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Voiculescu theorem, Sobolev lemma, and extensions of smooth algebras

Author: Xiaolu Wang
Journal: Bull. Amer. Math. Soc. 27 (1992), 292-297
MSC (2000): Primary 46L85; Secondary 19K33, 46M20, 47D25
MathSciNet review: 1161277
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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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Keywords: Fréchet algebra, smooth extension, operator ideal, $ \tau $-smooth completely positive map, quasi-central approximate identity
Article copyright: © Copyright 1992 American Mathematical Society

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