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 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.