The Gelfand Spectrum of a Noncommutative C*-algebra: a Topos-theoretic Approach

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 generalising Stone duality, maintains that the category of ‘pointfree spaces’ is the opposite of the cat­ egory of frames (i.e., complete lattices in which the meet distributes over arbitrary joins). One possible relationship between these two notions of space was unearthed by Banaschewski and Mulvey [“A globalisation of the Gelfand duality theorem”, Annals of Pure and Applied Logic 137, 62-103 (2006)], who proved a constructive version of Gelfand duality in which the Gelfand spectrum of a commutative C*-algebra comes out as a pointfree space. Being constructive, this result applies in arbitrary toposes (with natural numbers objects, so that internal C*-algebras can be defined). Earlier work by the first three authors [“A topos for algebraic quantum theory”, Communications in Mathematical Physics 291, 63-110 (2009)], shows how a noncom­ mutative 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 descrip­ tion of this internal spectrum, which in principle is a fibered pointfree space in the familiar topos Sets of sets and functions. However, we obtain the external spectrum as a fibered topological space in the usual sense. This leads to an explicit Gelfand transform, as well as to a topological reinterpretation of the Kochen-Specker Theo­ rem of quantum mechanics [“The problem of hidden variables in quantum mechanics”, Journal of Mathematics and Mechanics 17, 59-87 (1967)], which supplements the re­ markable topos-theoretic version of this theorem due to Butterfield and Isham [“A topos perspective on the Kochen-Specker theorem”, International Journal of Theoret­ ical Physics 37, 2669-2733 (1998)]. ‘ Computing Laboratory, Oxford University, Wolfson Building, Parks Road, Oxford OX1 3QD, U.K. Email: h e iu ie n @ c o m la b .o x .a c .u k . Supported by N.W.O. through a Rubicon grant. ^Radboud Universiteit Nijmegen, Institute for Mathematics, Astrophysics, and Particle Physics, Heyenda.alseweg 135, 6525 AJ NIJMEGEN, THE NETHERLANDS. Email: la n d s m a n @ m a th .r u .n l. *Radboud Universiteit Nijmegen, Institute for Computer and Information Science, Heyendaalseweg 135, 6525 AJ NIJMEGEN, THE NETHERLANDS. Email: s p i t t e r s @ c s . r u . n l . Supported by N.W.O. through the DIAMANT cluster. 3Radboud Universiteit Nijmegen, Institute for Mathematics, Astrophysics, and Particle Physics, Heyen­ daalseweg 135, 6525 AJ NIJMEGEN, THE NETHERLANDS. Email: s .w o l t e r s @ m a th .r u .n l . Supported by N.W.O. through project 613.000.811.