Noncommutative Semialgebraic sets and Associated Lifting Problems

TL;DR

This paper investigates lifting problems in noncommutative semialgebraic sets, demonstrating their projectivity and absolute retract properties, and establishing new results on the structure of C*-algebras related to these sets.

Contribution

It introduces new techniques for solving lifting problems involving soft polynomial relations and shows that various noncommutative semialgebraic sets are absolute retracts.

Findings

Certain noncommutative semialgebraic sets are absolute retracts.

The cone over any separable C*-algebra is an inductive limit of projective C*-algebras.

Projectivity of alternative noncommutative unit balls is established.

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.