The spectral theory of commutative C∗-algebras: the constructive spectrum

Banaschewski, Bernhard and Mulvey, Christopher J (2000) The spectral theory of commutative C∗-algebras: the constructive spectrum. Quaestiones Mathematicae, 23 (4). pp. 425-464. ISSN 1607-3606

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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 CE 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 CE. 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.

Item Type: Article
Keywords: LOCALE; COMMUTATIVE C*-ALGEBRA; SPECTRUM; PROPOSITIONAL GEOMETRIC THEORY; CONSTRUCTIVE; GELFAND DUALITY
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Depositing User: EPrints Services
Date Deposited: 06 Feb 2012 18:53
Last Modified: 17 Sep 2012 12:53
URI: http://sro.sussex.ac.uk/id/eprint/18795
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