Quillen property of real algebraic varieties

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Let I be a conjugation-invariant ideal in the complex polynomial ring with variables z₁,...,zₙ 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 Cⁿ 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 Cⁿ.