Noncommutative Semialgebraic sets and Associated Lifting Problems

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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.