No-go theorems for functorial localic spectra of noncommutative rings

TL;DR

The paper proves that certain functors assigning spectra to noncommutative rings are trivial on matrix algebras of size three or more, highlighting fundamental limitations in noncommutative spectral theory.

Contribution

It establishes no-go theorems showing that functorial localic spectra cannot distinguish noncommutative matrix algebras from trivial cases.

Findings

Functorial spectra are trivial on nxn-matrix algebras for n≥3.

Obstructions apply to Zariski, Stone, and Pierce spectra.

Spectral functors in categories other than locales are briefly considered.

Abstract

Any functor from the category of C*-algebras to the category of locales that assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on algebras of nxn-matrices for n at least 3. The same obstruction applies to the Zariski, Stone, and Pierce spectra. The possibility of spectra in categories other than that of locales is briefly discussed.