No-go theorems for functorial localic spectra of noncommutative rings
  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. 
Benno van den Berg, Chris Heunen
math.RA math.CT quant-ph