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

Abstract

Dixmier asked in 1967 whether the spectra of simple separable non-type-I $C^{\ast }$-algebras all have the same Borel structure. We solve the formulation using ordinary isomorphisms of quotient Borel $\sigma$-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 ${\omega }_2$ 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 ${\mathrm{SL}}_3(\mathbb{Z})$, there is no Borel-definable injection $\hat{B}\to \hat{A}$. 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 ${\omega }_1$ suffices for a positive answer. The finite-permutation Feferman--Levy model gives another positive answer for the full sigma-generated Borel structures. These comparisons concern the quotient structures themselves, without measure completion.

arXiv comment83 pages. Substantially expanded revision: distinguishes the quotient-measurable and Borel-definable formulations; adds ZFC independence via CH and the iterated Sacks model, results under choiceless hypotheses, and expanded proofs and literature comparisons