Christopher J. Mulvey

On the quantisation of points

In the study of quantales arising naturally in the context of -algebras, Gelfand quantales have emerged as providing the basic setting. In this paper, the problem of defining the concept of point of the spectrum of a -algebra A, which is one of the motivating examples of a Gelfand quantale, is considered. Intuitively, one feels that points should correspond to irreducible representations of A. Classically, the notions of topological and algebraic irreducibility of a representation are equivalent. In terms of quantales, the irreducible representations of a -algebra A are shown to be captured by the notion of an algebraically irreducible representation of the Gelfand quantale on an atomic orthocomplemented sup-lattice S, defined in terms of a homomorphism of Gelfand quantales to the Hilbert quantale of sup-preserving endomorphisms on S. This characterisation leads to a concept of point of an arbitrary Gelfand quantale Q as a map of Gelfand quantales into a Hilbert quantale , the inverse image homomorphism of which is an algebraically irreducible representation of Q on the atomic orthocomplemented sup-lattice S. The aptness of this definition of point is demonstrated by observing that in the case of locales it is exactly the classical notion of point, while the Hilbert quantale of an atomic orthocomplemented sup-lattice S has, up to equivalence, exactly one point. In this sense, the Hilbert quantale is considered to be a quantised version of the one-point space.

A globalisation of the Gelfand duality theorem

In this paper we bring together results from a series of previous papers to prove the constructive version of the Gelfand duality theorem in any Grothendieck topos E, obtaining a dual equivalence between the category of commutative C∗-algebras and the category of compact, completely regular locales in the topos E.

On the quantisation of spaces

Abstract In a previous paper (Pure Appl. Algebra 159 (2001) 231) a definition of point of a Gelfand quantale is given in terms of algebraically irreducible representations of the quantale on an atomic orthocomplemented sup-lattice. The definition yields the usual notion of point when applied to a locale viewed as a quantale and helps to establish that the spectrum Max A of a C ∗ -algebra A is an invariant of A. The current paper is concerned with finding a notion of spatiality in which, as intuitively expected, Max A is spatial, and for which there is an intimate connection with points in the above sense. Moreover, the approach taken relates naturally to that of Giles and Kummer in their article on non-commutative spaces (Indiana Math. J. 21 (1971) 91). Explicitly, an involutive unital quantale X is said to be spatial provided that it admits an algebraically strong right embedding into a discrete von Neumann quantale Q. It is proved that X is spatial if, and only if, it has enough points, and that, in this case, it is necessarily a Gelfand quantale. In particular, a locale is spatial as an involutive unital quantale precisely when it yields the topology of a classical topological space. A notion of quantal space is defined, and it is shown that any involutive unital quantale admits a spatialisation determined by its points, generalising that known for locales. In particular, any quantal space admits an underlying topological space. Finally, Max A is shown to be spatial, and to determine a canonical quantal space, as desired.

The spectral theory of commutative C∗-algebras: the constructive spectrum

This paper introduces the notion of a commutative C*-algebra in a Grothendieck topos E and subsequently that of the spectrum MFn A of A, presented as the locale determined by an appropriate propositional theory in the topos E which describes the basic properties of a multplicative linear functional on A. Further, the locale CE of complex numbers in the topos E is defined in a similar manner and some of its basic properties are established, such as its complete regularity and the compactness of the unit square in CE. Finally, it is shown that the locale MFn A is compact and completely regular, extending the classical result that the multiplicative linear functionals on a commutative C*-algebra form a compact Hausdorff space in the weak* topology.

The spectral theory of commutative C∗-algebras: The constructive Gelfand-Mazur theorem

It is shown, for a commutative C*-algebra in any Grothendieck topos E, that the locale MFn A of multiplicative linear functionals on A is isomorphic to the locale Max A of maximal ideals of A, extending the classical result that the space of C*-algebra homomorphisms from A to the field of complex numbers is isomorphic to the maximal ideal space of A, that is, the Gelfand-Mazur theorem, to the constructive context of any Grothendieck topos. The technique is to present Max A, in analogy with our earlier definition of MFn A, by means of a propositional theory which expresses ones natural intuition of the notion involved, and then to establish various properties, leading up to the final result, by formal reasoning within these theories.

A constructive proof of the Stone-Weierstrass theorem

Abstract A constructive version of the Stone-Weierstrass theorem is proved, allowing a globalisation of the Gelfand duality theorem to any Grothendieck topos to be established elsewhere.

A globalization of the Hahn-Banach theorem

The Hahn-Banach theorem is one of many fundamental results in functional analysis whose usefulness is diminished by their proof depending upon an application of the Axiom of Choice. The objection raised in this regard is not philosophical, but purely practical. When the existence of a functional has been established by this non-constructive means, the arbitrariness introduced restricts the information which can be extracted, impeding the discussion of points such as whether the functional can be shown to exist continuously with respect to a parameter [ 1, 15, 201, or equivariantly with respect to a group action [2, 333. This paper is concerned with outlining, starting from ideas which are already established, a context and a technique which allow the effects of this dependence on the Axiom of Choice to be avoided, reformulating the Hahn-Banach theorem in a way which may be applied equally to questions of continuity and of equivariance, to yield the information which is traditionally intended when applying the Hahn-Banach theorem. Before introducing the constructive context in which these ideas will be developed, consider the situations already mentioned, in which the arbitrariness of the existence of a functional established by the Axiom of Choice presents difficulties which need to be resolved. Firstly, take that of continuity in a parameter, which may be stated more precisely in the following way: consider a bundle B of seminormed spaces on a topological space X, together with a continuous mapping

The Stone–Čech compactification of locales, III

The Stone-Cech compactification of a locale L is shown to be obtained constructively by taking the Lindenbaum locale of the theory of almost prime completely regular filters on L. Modifying the theory by replacing the completely below relation by the strongly below relation yields instead the compact regular reflection, with corresponding results for the compact zero-dimensional reflection, indeed for any compactification of a locale.