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Let I be a conjugation-invariant ideal in the complex polynomial ring withvariables z_1,...,z_n and their conjugates. The ideal I has the Quillenproperty if every real valued, strictly positive polynomial on the real zeroset of I in C^n is a sum of hermitian squares modulo I. We first relate theQuillen property to the archimedean property from real algebra. Usinghereditary calculus, we then quantize and show that the Quillen propertyimplies the subnormality of commuting tuples of Hilbert space operatorssatisfying the identities in I. In the finite rank case we give a completegeometric characterization of when the identities in $I$ imply normality for acommuting tuple of matrices. This geometric interpretation provides simplemeans to refute Quillen's property of an ideal. We also generalize thesenotions and results from real algebraic sets to semi-algebraic sets in C^n.

Quillen property of real algebraic varieties