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

journal contribution

posted on 2023-06-07, 21:40 authored by Bernhard Banaschewski, Christopher J MulveyThis 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.

## History

## Publication status

- Published

## Journal

Quaestiones Mathematicae## ISSN

1607-3606## Publisher

Taylor and Francis## External DOI

## Issue

4## Volume

23## Page range

425-464## ISBN

1607-3606## Department affiliated with

- Mathematics Publications

## Full text available

- No

## Peer reviewed?

- Yes

## Legacy Posted Date

2012-02-06## Usage metrics

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