Jimmie D. Lawson

Spaces of maximal points

This paper shows that it is precisely the complete metrizable separable metric spaces that can be realized as the set of maximal points of an ω-continuous dcpo, where the set of maximal points is topologized with the relative Scott topology.

The spectral theory of distributive continuous lattices

In this paper various properties of the spectrum (i.e. the set of prime elements endowed with the hull-kernel topology) of a distributive continuous lattice are developed. It is shown that the spectrum is always a locally quasicompact sober space and conversely that the lattice of open sets of a locally quasicompact sober space is a continuous lattice. Algebraic lattices are a special subclass of continuous lattices and the special proper- ties of their spectra are treated. The concept of the patch topology is extended from algebraic lattices to continuous lattices, and necessary and sufficient conditions for its compactness are given. The spectral theory of lattices serves the purpose of representing a lattice L as a lattice of open sets of a topological space X. The spectral theory of rings and algebras practically reduces to this situation in view of the fact that for the most part one considers the lattice of ring (or algebra) ideals and then develops the spectral theory of that lattice. (The occasional complications due to the fact that ideal products are not intersections have been dealt with elsewhere, e.g. (4).) The lattice of all ring (or algebra) ideals forms a particular kind of continuous lattice, namely an algebraic lattice. It should be the case, however, that more general continuous lattices arise in the study of certain objects endowed with both an algebraic and a topological structure. Indeed the first author has shown in a seminar report using the concept of Pedersens ideal that the closed ideals of a C*-algebra always form a distributive continuous lattice with respect to intersection. How widely continuous lattices occur in such contexts is, at this point, a largely uncharted sea. We show that the spectrum of a distributive continuous lattice is a locally quasicompact sober space (see 2.6 for the definition of sobriety). This implies, e.g., that the space of closed two sided prime ideals of a C*-algebra is locally quasicompact in the hull-kernel topology. (This is usually proved for primitive ideals by different methods.) On the other hand, the question of what topological consequences follow

The Versatile Continuous Order

In this paper we survey some of the basic properties of continuously ordered sets, especially those properties that have led to their employment as the underlying structures for constructions in denotational semantics. The earlier sections concentrate on the order-theoretic aspects of continuously ordered sets and then specifically of domains. The last two sections are concerned with two natural topologies for sets with continuous orders, the Scott and Lawson topologies.

Karcher means and Karcher equations of positive definite operators

The Karcher or least-squares mean has recently become an important tool for the averaging and studying of positive definite matrices. In this paper we show that this mean extends, in its general weighted form, to the infinite-dimensional setting of positive operators on a Hilbert space and retains most of its attractive properties. The primary extension is via its characterization as the unique solution of the corresponding Karcher equation. We also introduce power means Pt in the infinite-dimensional setting and show that the Karcher mean is the strong limit of the monotonically decreasing family of power means as t → 0+. We show that each of these characterizations provide important insights about the Karcher mean.

Metric convexity of symmetric cones

AbstractIn this paper we introduce a general notion of a symmetric cone, valid for thefinite and infinite dimensional case, and prove that one can de duce the seminegativecurvature of the Thompson part metric in this general setting, along with standardinequalities familiar from operator theory. As a special case, we prove that everysymmetric cone from a JB-algebra satisfies a certain convexity property for theThompson part metric: the distance function between points evolving in time on twogeodesics is a convex function. This provides an affirmative answer to a question ofNeeb [22]. 1. IntroductionLet A be a unital C -algebra with identity e, and let A + be the set of positiveinvertible elements of A. It is known that A + is an open convex cone in the spaceH(A) of hermitian elements. The geometry of A + has been studied by several au-thors. One approach has been to endow A + with a natural Finsler structure and metricand use these for a substitute for the Riemannian geometry commonly considered infinite-dimensional examples. One particular focus in this geometry has been the studyof appropriate non-positive curvature properties. One prevalent notion of non-positivecurvature is a purely metric one, that of convexity of the metric. In [3], [4] and [9],Andruchow-Corach-Stojanoff and Corach-Porta-Recht haveshown the convexity of thedistance function along two distinct geodesics and its equivalence to the well-knownLoewner-Heinz inequality. In [22], Neeb established an appropriate differential geomet-ric notion of seminegative (equal non-positive) curvature for certain classes of Finslermanifolds.Our approach is somewhat different from either of the preceding. We replace thedifferential geometric structure by the structure of a symmetric space endowed with amidpoint operation and study seminegative curvature via convexity of the metric. In[16] we obtained the convexity of the metric for symmetric spaces with weaker metricassumptions than those enjoyed by the Finsler metric on A

MAXIMAL SUBSEMIGROUPS OF LIE GROUPS THAT ARE TOTAL

The major problem with which this paper is concerned is determining criteria that allow one to decide whether the subsemigroup generated by a subset B of a group G is all of G. Motivations for considering this problem arise from at least two sources. The first source concerns the program to develop a Lie theory of semigroups. Given a closed subsemigroup S of Lie group G, one defines the tangent object of S in the Lie algebra L(G) by

Symmetric spaces with convex metrics

Abstract We develop the basic theory of a general class of symmetric spaces, called lineated symmetric spaces, that satisfy the axioms of Loos together with an additional axiom that guarantees unique midpoints of symmetry. Our primary interest is the case that these symmetric spaces are Banach manifolds, in which case they exhibit an interesting geometric structure, and particularly in the metric case, where it is assumed the symmetric space carries a convex metric, an invariant complete metric contracting the square root function. One major result is that the distance function between points evolving over time on two geodesics is a convex function. Primary examples arise from involutive Banach-Lie groups (G,σ) admitting a polar decomposition G = P · K, where K is the subgroup fixed by σ and P is the associated symmetric set. We consider an appropriate notion of seminegative curvature for such symmetric spaces endowed with an invariant Finsler metric and prove that the corresponding length metric must be a convex metric. The preceding results provide a general framework for the interesting Finsler geometry of the space of positive Hermitian elements of a C*-algebra that has emerged in recent years.

D-completions and the d-topology

Abstract In this article we give a general categorical construction via reflection functors for various completions of T 0 -spaces subordinate to sobrification, with a particular emphasis on what we call the D -completion, a type of directed completion introduced by Wyler [O. Wyler, Dedekind complete posets and Scott topologies, in: B. Banaschewski, R.-E. Hoffmann (Eds.), Continuous Lattices Proceedings, Bremen 1979, in: Lecture Notes in Mathematics, vol. 871, Springer Verlag, 1981, pp. 384–389]. A key result is that all completions of a certain type are universal, hence unique (up to homeomorphism). We give a direct definition of the D -completion and develop its theory by introducing a variant of the Scott topology, which we call the d -topology. For partially ordered sets the D -completion turns out to be a natural dcpo-completion that generalizes the rounded ideal completion. In the latter part of the paper we consider settings in which the D -completion agrees with the sobrification respectively the closed ideal completion.

Computation on metric spaces via domain theory

Abstract The purpose of this paper is to survey recent approaches to realizing (or embedding) a Polish space as the set of maximal points of a continuous domain. Such realizations provide a convenient framework in which to model certain computational algorithms on the space and a useful alternate approach via the probabilistic power domain to measure theory and integraion on the space.

Weighted means and Karcher equations of positive operators

Significance As positive matrices and operators have gained increased prominence in theoretical, applied, and computational settings, finding appropriate methods for averaging them has become an important task. In recent years, the minimizer of the (weighted) sum of the distances (in an appropriately chosen metric) to the points to be averaged has been shown to exhibit many attractive features. In this paper, we extend most of these results to the infinite-dimensional setting, where the metric definition needs to be replaced by a solution shown to be unique of a corresponding equation called the Karcher equation. A multivariable weighted operator mean results that in many senses generalizes the geometric mean of a finite number of positive real numbers. The Karcher or least-squares mean has recently become an important tool for the averaging and study of positive definite matrices. In this paper, we show that this mean extends, in its general weighted form, to the infinite-dimensional setting of positive operators on a Hilbert space and retains most of its attractive properties. The primary extension is via its characterization as the unique solution of the corresponding Karcher equation. We also introduce power means in the infinite-dimensional setting and show that the Karcher mean is the strong limit of the monotonically decreasing family of power means as . We show each of these characterizations provide important insights about the Karcher mean.