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Formal Topology and Constructive Mathematics: the Gelfand and StoneYosida Representation Theorems
 Journal of Universal Computer Science
, 2005
"... Abstract. We present a constructive proof of the StoneYosida representation theorem for Riesz spaces motivated by considerations from formal topology. This theorem is used to derive a representation theorem for falgebras. In turn, this theorem implies the Gelfand representation theorem for C*alge ..."
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Abstract. We present a constructive proof of the StoneYosida representation theorem for Riesz spaces motivated by considerations from formal topology. This theorem is used to derive a representation theorem for falgebras. In turn, this theorem implies the Gelfand representation theorem for C*algebras of operators on Hilbert spaces as formulated by Bishop and Bridges. Our proof is shorter, clearer, and we avoid the use of approximate eigenvalues.
A Topos for Algebraic Quantum Theory
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2009
"... The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C ..."
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Cited by 8 (1 self)
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The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*algebra of observables A induces a topos T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum �(A) in T (A), which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topostheoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on �, and selfadjoint elements of A define continuous functions (more precisely, locale maps) from � to Scott’s interval domain. Noting that open subsets of �(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T (A). These results were inspired by the topostheoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
Constructive Completions of Ordered Sets, Groups and Fields
, 2003
"... In constructive mathematics it is of interest to consider a more general, but classically equivalent, notion of linear order, so called pseudoorder. ..."
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Cited by 2 (0 self)
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In constructive mathematics it is of interest to consider a more general, but classically equivalent, notion of linear order, so called pseudoorder.
Adjoints, absolute values and polar decompositions
 Journal of Operator Theory
"... Abstract. Various questions about adjoints, absolute values and polar decompositions of operators are addressed from a constructive point of view. The focus is on bilinear forms. Conditions are given for the existence of an adjoint, and a general notion of a polar decomposition is developed. The Rie ..."
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Abstract. Various questions about adjoints, absolute values and polar decompositions of operators are addressed from a constructive point of view. The focus is on bilinear forms. Conditions are given for the existence of an adjoint, and a general notion of a polar decomposition is developed. The Riesz representation theorem is proved without countable choice.
Real numbers and other completions
, 2007
"... A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an arbitrary metric space. The real numbers are characterized as a complete ..."
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A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an arbitrary metric space. The real numbers are characterized as a complete
Nonconstructive aspects of topology
, 2008
"... Inductively generated formal topologies Topology and choice principles It is wellknown that some basic theorems classical topology use (and require) the full Axiom of Choice: Tychonov’s Theorem (AC): If (Xi)i∈I is a family of (covering) compact spaces then the product topology is compact. Special c ..."
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Inductively generated formal topologies Topology and choice principles It is wellknown that some basic theorems classical topology use (and require) the full Axiom of Choice: Tychonov’s Theorem (AC): If (Xi)i∈I is a family of (covering) compact spaces then the product topology is compact. Special case: The Cantor space C = {0, 1} N is compact. i∈I
COMPUTABLE SETS: LOCATED AND OVERT LOCALES
, 2007
"... Abstract. What is a computable set? One may call a bounded subset of the plane computable if it can be drawn at any resolution on a computer screen. Using the constructive approach to computability one naturally considers totally bounded subsets of the plane. We connect this notion with notions intr ..."
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Abstract. What is a computable set? One may call a bounded subset of the plane computable if it can be drawn at any resolution on a computer screen. Using the constructive approach to computability one naturally considers totally bounded subsets of the plane. We connect this notion with notions introduced in other frameworks. A subset of a totally bounded set is again totally bounded iff it is located. 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 locale theory in a constructive, or topos theoretic, context. We show that the two notions are intimately connected. We propose a definition of located closed sublocale motivated by locatedness of subsets of metric spaces. A closed sublocale of a compact regular locale is located iff it is overt. Moreover, a closed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt. For Baire space metric locatedness corresponds to having a decidable positivity predicate. Finally, we show that the points of the Vietoris locale of a compact regular locale are precisely its compact overt sublocales. We work constructively, predicatively and avoid the use of the axiom of countable choice. Consequently, all are results are valid in any predicative topos. 1.
LOCATEDNESS AND OVERT SUBLOCALES
, 2009
"... 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 ..."
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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 set is Bishop compact iff it is located. We translate this result to formal topology. ‘Bishop compact ’ is translated as compact and overt. We propose a definition of located predicate on subspaces in formal topology. We call a sublocale located if it can be presented by a formal topology with a located predicate. We prove that a closed sublocale of a compact regular locale has a located predicate iff it is overt. Moreover, a Bishopclosed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt. Finally, we show by elementary methods that the points of the Vietoris locale of a compact regular locale are precisely its compact overt sublocales. We work constructively, predicatively and avoid the use of the axiom of countable choice. Consequently, all our results are valid in any predicative topos.