  Let I be a conjugation-invariant ideal in the complex polynomial ring with
variables z_1,...,z_n and their conjugates. The ideal I has the Quillen
property if every real valued, strictly positive polynomial on the real zero
set of I in C^n is a sum of hermitian squares modulo I. We first relate the
Quillen property to the archimedean property from real algebra. Using
hereditary calculus, we then quantize and show that the Quillen property
implies the subnormality of commuting tuples of Hilbert space operators
satisfying the identities in I. In the finite rank case we give a complete
geometric characterization of when the identities in $I$ imply normality for a
commuting tuple of matrices. This geometric interpretation provides simple
means to refute Quillen's property of an ideal. We also generalize these
notions and results from real algebraic sets to semi-algebraic sets in C^n.
