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

Abstract: We solve Dixmier's 1967 problem about spectra of simple $C<sup>*$-algebras. A function between spectra is called Borel-definable when it is induced by a Borel map between standard Borel spaces of unitary representations. Let $Γ=\mathrm{SL}<em>3(\mathbb{Z})$, let $K$ be its completion with respect to the congruence kernels modulo $2^n$, let $A$ be the canonical anticommutation relations (CAR) algebra, and let $B=C(K)\rtimes_rΓ$ be the reduced crossed product. The algebras $A$ and $B$ are simple, separable, unital, exact, and antiliminary; $A$ is nuclear, whereas $B$ is nonnuclear. There is no Borel-definable injection from the spectrum of $B$ to the spectrum of $A$. In fact, there is a probability measure on the pure-state space of $B$ such that every Borel lift of a Borel-definable function from the spectrum of $B$ to the spectrum of $A$ takes values in a single unitary-equivalence class almost everywhere. Answering a question of Simon Thomas, we also prove that, for every countable amenable group $H$, there is no Borel-definable injection from the unitary dual of the free group $F</em>\infty$ on infinitely many generators to the unitary dual of $H$. There is a fixed probability measure on a family of infinite-dimensional irreducible representations of $F_\infty$ such that every Borel lift of a Borel-definable function takes values in a single unitary-equivalence class almost everywhere. We also show that, in contrast, the spectra of any two separable nuclear non-type-I $C<sup>*$-algebras admit a Borel-definable bijection.