Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-29T11:10:37.665Z Has data issue: false hasContentIssue false

Constructive Gelfand duality for C*-algebras

Published online by Cambridge University Press:  01 September 2009

THIERRY COQUAND
Affiliation:
Computing Science Department at Göteborg University
BAS SPITTERS
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology e-mail: b.a.w.spitters@tue.nl

Abstract

We present a constructive proof of Gelfand duality for C*-algebras by reducing the problem to Gelfand duality for real C*-algebras.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Bis67]Bishop, E.Foundations of Constructive Analysis. (McGraw-Hill Book Co., 1967).Google Scholar
[BM97]Banaschewski, B. and Mulvey, C. J.A constructive proof of the Stone-Weierstrass theorem. J. Pure Appl. Algebra 116 (1–3) (1997), 2540. Special volume on the occasion of the 60th birthday of Professor Peter J. Freyd.CrossRefGoogle Scholar
[BM00a]Banaschewski, B. and Mulvey, C. J.The spectral theory of commutative C*-algebras: the constructive Gelfand–Mazur theorem. Quaest. Math., 23 (4) (2000), 465488.CrossRefGoogle Scholar
[BM00b]Banaschewski, B. and Mulvey, C. J.The spectral theory of commutative C*-algebras: the constructive spectrum. Quaest. Math., 23 (4) (2000), 425464.CrossRefGoogle Scholar
[BM06]Banaschewski, B. and Mulvey, C. J.A globalisation of the Gelfand duality theorem. Ann. Pure Appl. Logic., 137 (1–3) (2006), 62103.CrossRefGoogle Scholar
[CC00]Cederquist, J. and Coquand, T. Entailment relations and distributive lattices. In Logic Colloquium'98 (Prague), volume 13 of Lect. Notes Log., pages 127139. (Assoc. Symbol. Logic, Urbana, IL 2000).Google Scholar
[CLR01]Coste, M., Lombardi, H., and Roy, M.-F.Dynamical method in algebra: effective Nullstellensätze. Ann. Pure Appl. Logic 111 (3) (2001), 203256.CrossRefGoogle Scholar
[Coq05]Coquand, T.About Stone's notion of spectrum. J. Pure Appl. Alg., 197 (2005), 141158.CrossRefGoogle Scholar
[CS05]Coquand, T. and Spitters, B.Formal topology and constructive mathematics: the Gelfand and Stone-Yosida representation theorems. J. Universal Computer Science 11 (12) (2005), 19321944.Google Scholar
[HLS08]Heunen, C., Landsman, K. and Spitters, B. A topos for algebraic quantum theory. Submitted for publication, preprint available at http://arxiv.org/abs/0709.4364, (2008).Google Scholar
[Joh82]Johnstone, P. T.Stone Spaces (Cambridge University Press, 1982).Google Scholar
[Kri64]Krivine, J.-L.Anneaux préordonnés. J. Analyse Math. 12 (1964), 307326.CrossRefGoogle Scholar
[KV53]Kelley, J. L. and Vaught, R. L.The positive cone in banach algebras. Trans. Amer. Math. Soc. 74 (1953), 4455.CrossRefGoogle Scholar
[Mul03]Mulvey, C. J. On the geometry of choice. In Topological and algebraic structures in fuzzy sets, volume 20 of Trends Log. Stud. Log. Libr., pages 309336 (Kluwer Academic Publishers, 2003).CrossRefGoogle Scholar
[Wra80]Wraith, G. C. Intuitionistic algebra: some recent developments in topos theory. In Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pages 331337 (Helsinki, 1980). Acad. Sci. Fennica.Google Scholar