Solution to Dixmier's Problem about spectra of C*-algebras

Dixmier asked in 1967 whether the spectra of simple separable non-type-I C*-algebras all have the same Borel structure. We solve the formulation using ordinary isomorphisms of quotient Borel σ-algebras by proving that its universal assertion is independent of ZFC, relative to the consistency of ZFC. Under the continuum hypothesis, all separable non-type-I spectra are measurably isomorphic, without simplicity or unitality assumptions. In the model obtained by iterating Sacks forcing for ω₂ stages with countable supports over a GCH ground model, a fixed simple unital universal spectrum is not measurably isomorphic to the CAR spectrum. Neither consistency direction requires a large cardinal.

The Borel-definable formulation, which requires isomorphisms to admit Borel lifts between irreducible representations, has a negative answer in ZFC. For the CAR algebra A and a simple separable exact crossed product B associated to SL₃(ℤ), there is no Borel-definable injection from the spectrum of B to that of A. Nevertheless, all separable nuclear non-type-I spectra are Borel-definably isomorphic.

We also determine the answer under several choiceless hypotheses. The axiom of determinacy together with dependent choice gives a stronger negative conclusion: the spectrum of B does not even inject as a set into the spectrum of A. The symmetric Solovay model gives the same conclusion. Over ZF, a wellordering of the reals of order type ω₁ suffices for a positive answer. The finite-permutation Feferman–Lévy model gives another positive answer for the full sigma-generated Borel structures. These comparisons concern the quotient structures themselves, without measure completion.

GitHub repository: https://github.com/eulogica/dixmier-spectrum-problem