Bernhard Banaschewski

CAUCHY POINTS OF UNIFORM AND NEARNESS FRAMES

Abstract The notion of Cauchy point (= regular Cauchy filter) and the corresponding Cauchy spectrum, for a nearness frame (= uniform without the star-refinement condition) are investigated in various directions, including basic motivation, several functorial aspects, and the recognition of the Cauchy spectrum as the ordinary spectrum of the completion, after the unique existence of the latter is obtained as a central new result in this context.

BIFRAMES AND BISPACES

Abstract The concept of a biframe is introduced. Then the known dual adjunction between topological spaces and frames (i.e. local lattices) is extended to one between bispaces (i.e. bitopological spaces) and biframes. The largest duality contained in this dual adjunction defines the sober bispaces, which are also characterized in terms of the sober spaces. The topological and the frame-theoretic concepts of regularity, complete regularity and compactness are extended to bispaces and biframes respectively. For the bispaces these concepts are found to coincide with those introduced earlier by J.C. Kelly, E.P. Lane, S. Salbany and others. The Stone-Cech compactification (compact regular coreflection) of a biframe is constructed without the Axiom of Choice.

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 Function Ring Functor in Pointfree Topology

It is shown that the familiar existence of a left adjoint to the functor from the category of frames to the category of archimedean commutative f-rings with unit provided by the rings of pointfree continuous real-valued functions is already a consequence of a minimal amount of entirely obvious information, and this is then used to obtain unexpectedly simple proofs for a number of results concerning these function rings, along with their counterparts for the rings of integer-valued continuous functions in this setting. In addition, two different concrete descriptions are given for the left adjoint in question, one in terms of generators and relations motivated by the propositional theory of ℓ-ring homomorphisms into R, and the other based on a new notion of support specific to f-rings.

ON THE COMPLETION OF NEARNESS FRAMES

Abstract Several questions naturally arising from the unique existence of the completion of a nearness frame are investigated. In particular, the classical result that completion is a coreflection for uniform frames is extended to a substantially larger class of nearness frames but at the same time shown not to hold in general, and an analogue of this is established for the mere functoriality of the completion. Further, a natural variant of the notion of completion is studied, leading among other things to a completely new coreflection result.

Compactification and local connectedness of frames

Abstract Baboolal, D. and B. Banaschewski, Compactification and local connectedness of frames, Journal of Pure and Applied Algebra 70 (1991) 3-16. A classical result in the theory of Tychonoff spaces is that, for any such space X, its Stone-Tech compactification /IX is locally connected iff X is locally connected and pseudocompact. Since all concepts involved in this generalize from spaces to frames, it is natural to ask whether this result already holds for the latter, and the main purpose of this paper is to show this is indeed the case (Proposition 2.3). Further, for normal regular frames, we obtain the frame counterpart of an analogous result of Wallace in terms of a certain property of covers (Proposition 3.5). Finally, we establish a number of additional results concerning connectedness which seem to be of in- dependent interest. 0. Preliminaries Recall that a

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.

Uniform Completion In Pointfree Topology

Pointfree topology deals with certain complete lattices, called frames, which may be viewed as abstractly defined lattices of open sets, sufficiently resembling the concrete lattices of this kind that arise from topological spaces to make the treatment of a variety of topological questions possible. It turns out that a remarkable number of topological facts derive from results in this pointfree setting while the proofs of the latter are often more suggestive and transparent than those of their classical counterparts. But there is a deeper aspect of frames which endows them with a very specific significance: various topological spaces classically associated with other entities (such as several types of rings, or Banach spaces, or lattices) are actually the spectra of appropriate frames which themselves require weaker logical foundations for the proofs of their basic properties than those needed for the actual spaces but which can still serve much the same purposes as the spaces in question. In this way, pointfree topology acquires an autonomous role and appears as more fundamental than classical topology.

Variants of openness

Algebraic conditions on frame homomorphisms representing various types of openness requirements on continuous maps are investigated. It turns out that several of these can be expressed in terms of formulas involving pseudocomplements. A full classification of the latter is presented which shows that they group into five equivalence classes and establishes the logical connections between them. Among the relation of our algebraic conditions to continuous maps between topological spaces, we establish that the coincidence of the algebraic and topological notion of openness is equivalent to the separation axiomTD for the domain space.

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.