## 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