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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Noncommutative semialgebraic sets and associated lifting problems
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by Terry A. Loring and Tatiana Shulman PDF
Trans. Amer. Math. Soc. 364 (2012), 721-744 Request permission

Abstract:

We solve a class of lifting problems involving approximate polynomial relations (soft polynomial relations). Various associated $C^{*}$-algebras are therefore projective. The technical lemma we need is a new manifestation of Akemann and Pedersen’s discovery of the norm adjusting power of quasi-central approximate units.

A projective $C^{*}$-algebra is the analog of an absolute retract. Thus we can say that various noncommutative semialgebraic sets turn out to be absolute retracts. In particular we show that a noncommutative absolute retract results from the intersection of the approximate locus of a noncommutative homogeneous polynomial with the noncommutative unit ball. By unit ball we are referring to the $C^{*}$-algebra of the universal row contraction. We show that various alternative noncommutative unit balls are also projective.

Sufficiently many $C^{*}$-algebras are now known to be projective so that we are able to show that the cone over any separable $C^{*}$-algebra is the inductive limit of $C^{*}$-algebras that are projective.

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Additional Information
  • Terry A. Loring
  • Affiliation: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131
  • Tatiana Shulman
  • Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
  • Address at time of publication: Department of Mathematics, Siena College, 515 Loudon Road, Loudonville, New York 12211
  • MR Author ID: 684365
  • Received by editor(s): August 23, 2009
  • Received by editor(s) in revised form: January 29, 2010, and February 10, 2010
  • Published electronically: October 5, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 721-744
  • MSC (2010): Primary 46L05, 47B99
  • DOI: https://doi.org/10.1090/S0002-9947-2011-05313-4
  • MathSciNet review: 2846350