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

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

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: heunen@comlab.ox.ac.uk)
NICOLAAS P. LANDSMAN
Affiliation:
Institute for Mathematics, Astrophysics, and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands (email: landsman@math.ru.nl)
BAS SPITTERS
Affiliation:
Institute for Computer and Information Science, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands (email: spitters@cs.ru.nl)
SANDER WOLTERS
Affiliation:
Institute for Mathematics, Astrophysics, and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands (email: s.wolters@math.ru.nl)
*
✉For correspondence; e-mail: heunen@comlab.ox.ac.uk
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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.

References

[1]Akemann, C. A., ‘A Gelfand representation theory for C *-algebras’, Pacific J. Math. 39 (1971), 1–11.CrossRefGoogle Scholar
[2]Banaschewski, B. and Mulvey, C. J., ‘The spectral theory of commutative C*-algebras: the constructive Gelfand–Mazur theorem’, Quaest. Math. 23 (2000), 465–488.CrossRefGoogle Scholar
[3]Banaschewski, B. and Mulvey, C. J., ‘The spectral theory of commutative C*-algebras: the constructive spectrum’, Quaest. Math. 23 (2000), 425–464.CrossRefGoogle Scholar
[4]Banaschewski, B. and Mulvey, C. J., ‘A globalisation of the Gelfand duality theorem’, Ann. Pure Appl. Logic 137 (2006), 62–103.CrossRefGoogle Scholar
[5]Bohr, N., ‘Discussion with Einstein on epistemological problems in atomic physics’, in: Albert Einstein: Philosopher–Scientist (ed. Schilpp, P. A.) (Open Court, La Salle, IL, 1969),pp. 201–241.Google Scholar
[6]Borceux, F., Handbook of Categorical Algebra. 3. Categories of Sheaves (Cambridge University Press, Cambridge, 1994).Google Scholar
[7]Butterfield, J. and Isham, C. J., ‘A topos perspective on the Kochen–Specker theorem: I. Quantum states as generalized valuations’, Internat. J. Theoret. Phys. 37 (1998), 2669–2733.Google Scholar
[8]Caspers, M., Heunen, C., Landsman, N. P. and Spitters, B., ‘Intuitionistic quantum logic of an n-level system’, Found. Phys. 39 (2009), 731–759.CrossRefGoogle Scholar
[9]Choukri, R., Illoussamen, E. H. and Runde, V., ‘Gelfand theory for non-commutative Banach algebras’, Q. J. Math. 53 (2002), 161–172.CrossRefGoogle Scholar
[10]Connes, A., Noncommutative Geometry (Academic Press, San Diego, CA, 1994).Google Scholar
[11]Connes, A. and Marcolli, M., Noncommutative Geometry, Quantum Fields and Motives (American Mathematical Society, Providence, RI, 2008).Google Scholar
[12]Coquand, T., ‘About Stone’s notion of spectrum’, J. Pure Appl. Algebra 197 (2005), 141–158.CrossRefGoogle Scholar
[13]Coquand, T. and Spitters, B., ‘Constructive Gelfand duality for C *-algebras’, Math. Proc. Cambridge Philos. Soc. 147 (2009), 339–344.CrossRefGoogle Scholar
[14]Dauns, J. and Hofmann, K. H., Representation of Rings by Sections, Memoirs of the American Mathematical Society, 83 (American Mathematical Society, Providence, RI, 1968).CrossRefGoogle Scholar
[15]Fell, J. M. G. and Doran, R. S., Representations of *-algebras, Locally Compact Groups, and Banach *-algebraic Bundles, Vol. 2 (Academic Press, Boston, MA, 1988).Google Scholar
[16]Fourman, M. P. and Scott, D. S., ‘Sheaves and logic’, in: Applications of Sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Univ. Durham, Durham, 1977), Lecture Notes in Mathematics, 753 (Springer, Berlin, 1979), pp. 302–401.Google Scholar
[17]Goldblatt, R., Topoi, The Categorical Analysis of Logic (North-Holland, Amsterdam, 1984).Google Scholar
[18]Heunen, C., Landsman, N. P. and Spitters, B., ‘A topos for algebraic quantum theory’, Comm. Math. Phys. 291 (2009), 63–110.CrossRefGoogle Scholar
[19]Johnstone, P. T., Stone Spaces (Cambridge University Press, Cambridge, 1982).Google Scholar
[20]Johnstone, P. T., Sketches of an Elephant: a Topos Theory Compendium, Vol. 1 (Clarendon Press, Oxford, 2002).Google Scholar
[21]Johnstone, P. T., Sketches of an Elephant: a Topos Theory Compendium, Vol. 2 (Clarendon Press, Oxford, 2002).Google Scholar
[22]Joyal, A. and Tierney, M., An Extension of the Galois Theory of Grothendieck, Vol. 51, Memoirs of the American Mathematical Society (American Mathematical Society, Providence, RI, 1984).Google Scholar
[23]Kochen, S. and Specker, E. P., ‘The problem of hidden variables in quantum mechanics’, J. Math. Mech. 17 (1967), 59–87.Google Scholar
[24]Kruszyński, P. and Woronowicz, S. L., ‘A noncommutative Gelfand–Naimark theorem’, J. Operator Theory 8 (1982), 361–389.Google Scholar
[25]Landsman, N. P., ‘Poisson spaces with a transition probability’, Rev. Math. Phys. 9(1) (1997), 29–57.CrossRefGoogle Scholar
[26]Landsman, N. P., Mathematical Topics between Classical and Quantum Mechanics (Springer, Berlin, 1998).CrossRefGoogle Scholar
[27]Landsman, N. P., ‘Between classical and quantum’, in: Handbook of Philosophy of Science, Philosophy of Physics, 2 (eds. Earman, J. and Butterfield, J.) (Elsevier, Amsterdam, 2007),pp. 417–553.Google Scholar
[28]Lane, S. Mac and Moerdijk, I., Sheaves in Geometry and Logic (Springer, Berlin, 1992).Google Scholar
[29]Mulvey, C. J., ‘A noncommutative Gelfand–Naimark theorem’, Math. Proc. Cambridge Philos. Soc. 88 (1980), 425–428.CrossRefGoogle Scholar
[30]Redhead, M. L. G., Incompleteness, Nonlocality and Realism: a Prolegomenon to the Philosophy of Quantum Mechanics (Clarendon Press, Oxford, 1987).Google Scholar
[31]Shultz, F. W., ‘Pure states as a dual object for C *-algebras’, Comm. Math. Phys. 82 (1981/82), 497–509.CrossRefGoogle Scholar
[32]Spitters, B., ‘The space of measurement outcomes as a spectrum for non-commutative algebras’, EPTCS, doi: http://dx.doi.org/10.4204/EPTCS.26.12.CrossRefGoogle Scholar
[33]van den Berg, B. and Heunen, C., ‘Noncommutativity as a colimit’. Appl. Categ. Structures, doi: 10.1007/S10485-011-9246-3.Google Scholar
[34]Vickers, S., Locales and Toposes as Spaces, Handbook of Spatial Logics, 8 (eds. Aiello, M., Pratt-Hartmann, I. E. and van Benthem, J. F. A. K.) (Springer, Berlin, 2007), pp. 429–496.Google Scholar
[35]Wolters, S., ‘Contravariant vs covariant quantum logic: a comparison of two topos-theoretic approaches to quantum theory’, arXiv:1010.2031.Google Scholar