Joan Wick Pelletier

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.

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.

On quantales and spectra of C*-algebras

We study properties of the quantale spectrum Max A of an arbitrary unital C*-algebra A. In particular we show that the spatialization of Max A with respect to one of the notions of spatiality in the literature yields the locale of closed ideals of A when A is commutative. We study under general conditions functors with this property, in addition requiring that colimits be preserved, and we conclude in this case that the spectrum of A necessarily coincides with the locale of closed ideals of the commutative reflection of A. Finally, we address functorial properties of Max, namely studying (non-)preservation of limits and colimits. Although Max  is not an equivalence of categories, therefore not providing a direct generalization of Gelfand duality to the noncommutative case, it is a faithful complete invariant of unital C*-algebras.

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

Interpolation functors and duality

Preliminaries.- The real method.- The complex method.- Categorical notions.- Finite dimensional doolittle diagrams.- Kan extensions.- Duality.- More about duality.- The classical methods from a categorical viewpoint.

Locales in functional analysis

Abstract Locales, as a generalization of the notion of topological space, play a crucial role in allowing theorems of functional analysis which classically depend on the Axiom of Choice to be suitably reformulated and proved in the intuitionistic context of a Grothendieck topos. The manner in which locales arise and the role they play are discussed in connection with the Hahn-Banach and Gelfand duality theorems.

An extension of the Aronszajn-Gagliardo theorem

In 1965 Aronszajn and Gagliardo [l] proved that any interpolation space of a given Banach couple could be realized as the value of a minimal or maximal interpolation functor on the category of all Banach couples. This result was, paradoxically, both considered a basic result of the theory and ignored. It is only recently that attention has once again been focussed on it in view of the discovery by Brudnyi-Krugljak [3] and Janson [8] that there exists a strong connection between the important methods of interpolation and the Aronszajn-Gagliardo theorem. For example, Janson has shown that many interpolation functors, including the real and complex methods, are minimal or maximal extensions from a single Banach couple, hence, ‘Aronszajn-Gagliardo functors’. While the category of Banach spaces has been and continues to be the most studied setting for interpolation theory, applications have indicated the desirability and need to have a theory of interpolation in other settings. A particular need exists for the category of quasi-normed spaces, the setting of the classical Marcinkiewicz interpolation theorem of 1939 [2], in order to obtain a full generalization of this theorem. Accordingly, interpolation methods have been studied in this category by KrCe [ 131, Holmstedt [7], and Sagher [ 171, and real methods of interpolation have been defined there. Further prompted by applications to approximation theory, Peetre and Sparr [16] have developed a theory of interpolation for quasi-normed abelian groups and normed abelian groups. Categories of weaker linear structures,

Tensor norms and operators in the category of Banach spaces

The theory of dual functors in the category of Banach spaces is applied to the study of tensor norms in the sense of Grothendieck. The dual functors of the tensor norms arising from the projective and inductive tensor product as well as from more general tensor norms, such as the norms dp of Saphar, are identified as various spaces of operators, which include p-integral and absolutely p-summing operators. Properties of these operators are then easily derived by categorical means. Applications of the methods provide simplified proofs of composition theorems and the characterization of dual spaces of type (L).