11 found
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  1. Bohrification of operator algebras and quantum logic.Chris Heunen, Nicolaas P. Landsman & Bas Spitters - 2012 - Synthese 186 (3):719 - 752.
    Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hubert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families (...)
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  2.  24
    Constructive algebraic integration theory.Bas Spitters - 2006 - Annals of Pure and Applied Logic 137 (1-3):380-390.
    For a long time people have been trying to develop probability theory starting from ‘finite’ events rather than collections of infinite events. In this way one can find natural replacements for measurable sets and integrable functions, but measurable functions seemed to be more difficult to find. We present a solution. Moreover, our results are constructive.
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  3. Intuitionistic Quantum Logic of an n-level System.Martijn Caspers, Chris Heunen, Nicolaas P. Landsman & Bas Spitters - 2009 - Foundations of Physics 39 (7):731-759.
    A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the (...)
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  4. A Constructive View on Ergodic Theorems.Bas Spitters - 2006 - Journal of Symbolic Logic 71 (2):611 - 623.
    Let T be a positive L₁-L∞ contraction. We prove that the following statements are equivalent in constructive mathematics. (1) The projection in L₂ on the space of invariant functions exists: (2) The sequence (Tⁿ)n∈N Cesáro-converges in the L₂ norm: (3) The sequence (Tⁿ)n∈N Cesáro-converges almost everywhere. Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem. As a corollary we obtain a constructive ergodic theorem for ergodic measure-preserving transformations. This answers a question (...)
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  5. The principle of general tovariance.Chris Heunen, Klaas Landsman & Bas Spitters - unknown
    We tentatively propose two guiding principles for the construction of theories of physics, which should be satisfied by a possible future theory of quantum gravity. These principles are inspired by those that led Einstein to his theory of general relativity, viz. his principle of general covariance and his equivalence principle, as well as by the two mysterious dogmas of Bohr's interpretation of quantum mechanics, i.e. his doctrine of classical concepts and his principle of complementarity. An appropriate mathematical language for combining (...)
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  6.  26
    Corrigendum to: 'A Constructive View on Ergodic Theorems'.Bas Spitters - 2006 - Journal of Symbolic Logic 71 (4):1431 - 1432.
  7.  41
    A constructive proof of the Peter‐Weyl theorem.Thierry Coquand & Bas Spitters - 2005 - Mathematical Logic Quarterly 51 (4):351-359.
    We present a new and constructive proof of the Peter-Weyl theorem on the representations of compact groups. We use the Gelfand representation theorem for commutative C*-algebras to give a proof which may be seen as a direct generalization of Burnside's algorithm [3]. This algorithm computes the characters of a finite group. We use this proof as a basis for a constructive proof in the style of Bishop. In fact, the present theory of compact groups may be seen as a natural (...)
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  8.  31
    Metric complements of overt closed sets.Thierry Coquand, Erik Palmgren & Bas Spitters - 2011 - Mathematical Logic Quarterly 57 (4):373-378.
    We show that the set of points of an overt closed subspace of a metric completion of a Bishop-locally compact metric space is located. Consequently, if the subspace is, moreover, compact, then its collection of points is Bishop-compact. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
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  9.  39
    Locatedness and overt sublocales.Bas Spitters - 2010 - Annals of Pure and Applied Logic 162 (1):36-54.
    Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in constructive locale theory. We show that the two notions are intimately connected.Bishop defines a metric space to be compact if it is complete and totally bounded. A subset of a totally bounded set is again totally bounded iff it is located. So a closed subset of a Bishop compact (...)
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  10.  17
    Located Operators.Bas Spitters - 2002 - Mathematical Logic Quarterly 48 (S1):107-122.
    We study operators with located graph in Bishop-style constructive mathematics. It is shown that a bounded operator has an adjoint if and only if its graph is located. Locatedness of the graph is a necessary and sufficient condition for an unbounded normal operator to have a spectral decomposition. These results suggest that located operators are the right generalization of bounded operators with an adjoint.
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  11.  51
    The Space of Measurement Outcomes as a Spectral Invariant for Non-Commutative Algebras.Bas Spitters - 2012 - Foundations of Physics 42 (7):896-908.
    The recently developed technique of Bohrification associates to a (unital) C*-algebra Athe Kripke model, a presheaf topos, of its classical contexts;in this Kripke model a commutative C*-algebra, called the Bohrification of A;the spectrum of the Bohrification as a locale internal in the Kripke model. We propose this locale, the ‘state space’, as a (n intuitionistic) logic of the physical system whose observable algebra is A.We compute a site which externally captures this locale and find that externally its points may be (...)
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