The spectral theory of commutative C^*-algebras: The constructive spectrum

Abstract

This paper introduces the notion of a commutative C^*-algebra in a Grothendieck topos E and subsequently that of the spectrum MFn A of A, presented as the locale determined by an appropriate propositional theory in the topos E which describes the basic properties of a multplicative linear functional on A. Further, the locale C_E of complex numbers in the topos E is defined in a similar manner and some of its basic properties are established, such as its complete regularity and the compactness of the unit square in C_E. Finally, it is shown that the locale MFn A is compact and completely regular, extending the classical result that the multiplicative linear functionals on a commutative C^*-algebra form a compact Hausdorff space in the weak^* topology.