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

We solve Dixmier's 1967 problem about spectra of simple $C^*$-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 $Γ=\mathrm{SL}_3(\mathbb{Z})$, let $K$ be its completion with respect to the co