The Gelfand spectrum of a noncommutative C*-algebra: a topos-theoretic approach

We compare two influential ways of defining a generalized notion of space. The first, inspired by Gelfand duality, states that the category of ‘noncommutative spaces’ is the opposite of the category of C*-algebras. The second, loosely generalizing Stone duality, maintains that the category of ‘point-free spaces’ is the opposite of the category of frames (that is, complete lattices in which the meet distributes over arbitrary joins). Earlier work by the first three authors shows how a noncommutative C*-algebra gives rise to a commutative one internal to a certain sheaf topos. The latter, then, has a constructive Gelfand spectrum, also internal to the topos in question. After a brief review of this work, we compute the so-called external description of this internal spectrum, which in principle is a fibred point-free space in the familiar topos of sets and functions. However, we obtain the external spectrum as a fibred topological space in the usual sense. This leads to an explicit Gelfand transform, as well as to a topological reinterpretation of the Kochen–Specker theorem of quantum mechanics.