Noncommutative Semialgebraic sets and Associated Lifting Problems
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 a noncommutative absolute retract results from the intersection of the approximate locus of a homogeneous polynomial with the noncommutative unit ball. By unit ball we are referring the C*-algebra of the universal row contraction. We show projectivity of alternative noncommutative unit balls. Sufficiently many C*-algebras are now known to be projective that we are able to show that the cone over any separable C*-algebra is the inductive limit of C*-algebras that are projective.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2009
- DOI:
- 10.48550/arXiv.0907.2618
- arXiv:
- arXiv:0907.2618
- Bibcode:
- 2009arXiv0907.2618L
- Keywords:
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- Mathematics - Operator Algebras;
- 46L05
- E-Print:
- 23 pages. Completely new section: Cones are Limits of Projective C*-Algebras