The spectral theory of commutative C^∗-algebras: The constructive Gelfand-Mazur theorem

Abstract

It is shown, for a commutative C^∗-algebra in any Grothendieck topos E, that the locale MFn A of multiplicative linear functionals on A is isomorphic to the locale Max A of maximal ideals of A, extending the classical result that the space of C^∗-algebra homomorphisms from A to the field of complex numbers is isomorphic to the maximal ideal space of A, that is, the Gelfand-Mazur theorem, to the constructive context of any Grothendieck topos. The technique is to present Max A, in analogy with our earlier definition of MFn A, by means of a propositional theory which expresses one's natural intuition of the notion involved, and then to establish various properties, leading up to the final result, by formal reasoning within these theories.

Keywords:

• LOCALE
• COMMUTATIVE C*-ALGEBRA
• SPECTRUM
• PROPOSITIONAL GEOMETRIC THEORY
• CONSTRUCTIVE
• GELFAND DUALITY
• GELFAND-MAZUR THEOREM