Abstract. We solve a class of lifting problems involving approximate polynomial relations (“softened polynomial relations”). Various associated C ∗-algebras are therefore projective. The technical lemma we need is new manifestation of Akemann and Pedersen’s discovery of the norm adjusting power of quasi-central approximate units. The unitization of projective C ∗-algebras 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
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