No-go theorems for functorial localic spectra of noncommutative rings

Any functor from the category of C*-algebras to the category of locales which assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on algebras of n-by-n matrices for n ≥ 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. A recent article [7] by Reyes shows that any functor Cstar → Top which assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on the matrix algebras Mn(C) for n ≥ 3. The introduction of that paper suggests that this might be because the spectrum ought to be a “space without points”. This suggests that a functorial extension which is nontrivial on the matrix algebras Mn(C) for n ≥ 3 might still be possible in locale theory, where there is a sensible notion of pointfree space. Unfortunately, this is not the case: we show that any functor Cstar → Loc which assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on Mn(C) for n ≥ 3. The same obstruction applies to the Zariski, Stone, and Pierce spectra. 1 Locale-theoretic preliminaries Locales and topological spaces are closely related apart from a few subtle differences. One of the most important is that in these categories limits are, in general, computed differently. Initially one might hope that for this reason Reyes’ result does not apply to locales, but it turns out that it does. The key observation is that, although limits in spaces and locales differ in general, they coincide for those spaces (locales) that arise as spectra. Proposition 1. Both compact regular and compact completely regular locales are closed under limits in Loc. Proof. The product of compact (completely) regular locales is again (completely) regular and compact [4, III.1.6, III.1.7, IV.1.5]. The equalizer of f, g:A→ B is a closed sublocale of A whenever B is (completely) regular [4, III.1.3] and a closed sublocale of a compact (completely) regular locale is again (completely) regular and compact [4, III.1.2, IV.1.5]. ∗Technische Universität Darmstadt, Fachbereich Mathematik. †Oxford University Computing Laboratory, supported by the Netherlands Organisation for Scientific Research (NWO).