Universality of the CAR separation among simple exact nonnuclear algebras

Determine whether every simple exact nonnuclear C*-algebra has a definable spectrum strictly above the definable spectrum of the CAR algebra.

Background

The paper proves that one specific simple, exact, nonnuclear crossed product has definable spectrum strictly above the CAR degree. This demonstrates existence of a separation but does not establish that all simple exact nonnuclear C*-algebras exhibit such a separation.

The question asks whether the strict increase in definable complexity is universal throughout the class of simple exact nonnuclear C*-algebras.

References

Does every simple exact nonnuclear $C*$-algebra have a definable spectrum strictly above the CAR degree? Theorem~A supplies one such algebra, but does not establish this universal reading of Farah's Problem~9.

— Solution to Dixmier's Problem about spectra of C*-algebras  (2609.26319 - Lupini, 22 Sep 2026) in Question 6.1(iii), Section 6, subsection 'Limits of the obstruction'