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^*-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 B̂→Â. 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--Levy model gives another positive answer for the full sigma-generated Borel structures. These comparisons concern the quotient structures themselves, without measure completion.

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BibTeX RIS

BibTeX

@misc{lupini2026solution,
  title = {{Solution to Dixmier's Problem about spectra of C*-algebras}},
  author = {Martino Lupini},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.26319},
  eprint = {2609.26319},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26319},
}

RIS

TY  - UNPB
TI  - Solution to Dixmier's Problem about spectra of C*-algebras
AU  - Martino Lupini
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.26319
UR  - https://arxiv.org/abs/2609.26319
AB  - 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 ω_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 SL_3(ℤ), there is no Borel-definable injection B̂→Â. 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 ω_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.
ER  -

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