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THE GELFAND SPECTRUM OF A NONCOMMUTATIVE C*-ALGEBRA: A TOPOS-THEORETIC APPROACH

Part of: Proof theory and constructive mathematics Selfadjoint operator algebras Linear spaces and algebras of operators Distributive lattices Boolean algebras (Boolean rings)

Published online by Cambridge University Press:  08 April 2011

CHRIS HEUNEN ,
NICOLAAS P. LANDSMAN ,
BAS SPITTERS  and
SANDER WOLTERS
CHRIS HEUNEN*
Affiliation:
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK (email: [email protected])
NICOLAAS P. LANDSMAN
Affiliation:
Institute for Mathematics, Astrophysics, and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands (email: [email protected])
BAS SPITTERS
Affiliation:
Institute for Computer and Information Science, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands (email: [email protected])
SANDER WOLTERS
Affiliation:
Institute for Mathematics, Astrophysics, and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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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.

Keywords

Gelfand spectrumnoncommutative C*-algebratoposlocale

MSC classification

Secondary: 46L85: Noncommutative topology 47L40: Limit algebras, subalgebras of $C^*$-algebras 06D22: Frames, locales 03F55: Intuitionistic mathematics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 1 , February 2011 , pp. 39 - 52
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

Heunen was supported by the Netherlands Organisation for Scientific Research through a Rubicon grant; Spitters was supported by the Netherlands Organisation for Scientific Research through the diamant cluster; Wolters was supported by the Netherlands Organisation for Scientific Research through project 613.000.811.

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