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Transactions of the American Mathematical Society

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



Noncommutative semialgebraic sets and associated lifting problems

Authors: Terry A. Loring and Tatiana Shulman
Journal: Trans. Amer. Math. Soc. 364 (2012), 721-744
MSC (2010): Primary 46L05, 47B99
Published electronically: October 5, 2011
MathSciNet review: 2846350
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Abstract | References | Similar Articles | Additional Information


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

Keywords: $C^{*}$-algebra, relation, projectivity, row contraction, noncommutative star-polynomial, lifting.
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.