On Order Bounded Subsets of Locally Solid Riesz Spaces

In a topological Riesz space there are two types of bounded subsets: order bounded subsets and topologically bounded subsets. It is natural to ask (1) whether an order bounded subset is topologically bounded and (2) whether a topologically bounded subset is order bounded. A classical result gives a partial answer to (1) by saying that an order bounded subset of a locally solid Riesz space is topologically bounded. This paper attempts to further investigate these two questions. In particular, we show that (i) there exists a non-locally solid topological Riesz space in which every order bounded subset is topologically bounded; (ii) if a topological Riesz space is not locally solid, an order bounded subset need not be topologically bounded; (iii) a topologically bounded subset need not be order bounded even in a locally convex-solid Riesz space. Next, we show that (iv) if a locally solid Riesz space has an order bounded topological neighborhood of zero, then every topologically bounded subset is order bounded; (v) however, a locally convex-solid Riesz space may not possess an order bounded topological neighborhood of zero even if every topologically bounded subset is order bounded; (vi) a pseudometrizable locally solid Riesz space need not have an order bounded topological neighborhood of zero. In addition, we give some results about the relationship between order bounded subsets and positive homogeneous operators.     

On the variety of Riesz spaces

Finitely generated linearly ordered Riesz spaces are described, leading to a proof that the variety of Riesz spaces is generated as a quasivariety by the Riesz space ? of real numbers. The finitely generated Riesz spaces are also described: they are the subalgebras of real-valued function spaces on root systems of finite height.     

Representations of Archimedean Riesz spaces by continuous functions

A brief survey of representations of Archimedean Riesz spaces in spaces of continuous extended real-valued functions, together with an example of their use in proving results about Riesz spaces     

Strict and Mackey topologies and tight vector measures

Let B() denote the Banach algebra of all bounded Borel measurable complex functions defined on a topological Hausdor? space X, and B_o() stand for the ideal of B() consisting of all functions vanishing at infinity. Then B() is a faithful Banach left B_o()-module and the strict topology β on B() induced by B_o() is a mixed topology. For a sequentially complete locally convex Hausdor? space (E, ξ), we study the relationship between vector measures m : → E and the corresponding continuous integration operators T_m : B() → E. It is shown that a measure m : → E is countably additive tight if and only if the corresponding integration operator T_m is (η, ξ)-continuous, where η denotes the infimum of the strict topology β and the Mackey topology τ (B(), ca()). If, in particular, E is a Banach space, it is shown that m is countably additive tight if and only if Tm(absconv(U ∪ W)) is relatively weakly compact in E for some τ (B(), ca())-neighborhood U of 0 and some β-neighborhood W of 0 in B(). As an application, we prove a Nikodym type convergence theorem for countably additive tight vector measures.     

Jordan Structures in Harmonic Functions and Fourier Algebras on Homogeneous Spaces

We study the Jordan structures and geometry of bounded matrix-valued harmonic functions on a homogeneous space and their analogue, the harmonic functionals, in the setting of Fourier algebras of homogeneous spaces.Supported by EPSRC grant GR/G91182 and NSERC grant 7679.     

Pointfree prime representation of real Riesz maps

In classical topology it is proved, nonconstructively, that for a topological space X, every bounded Riesz map ϕ in C(X) is of the form for a point x ∈ X. In this paper our main objective is to give the pointfree version of this result. In fact, we constructively represent each real Riesz map on a compact frame M by prime elements. Received March 23, 2004; accepted in final form May 14, 2005.     

2-Universally complete Riesz spaces and inverse-closed Riesz spaces

In this paper we show mainly two results about uniformly closed Riesz subspaces of ?^X containing the constant functions. First, for such a Riesz subspace E, we solve the problem of determining the properties that a real continuous functiondefined on a proper open interval of ?should have in order that the conditions “E is closed under composition with ” and “E is closed under inversion in X” become equivalent. The second result, reformulated in the more general frame of the Archimedean Riesz spaces with weak order unit e, establishes that E (e-uniformly complete and e-semisimple) is closed under inversion in C(Spec E) if and only if E is 2-universally e-complete.     

Disjointness in Partially Ordered Vector Spaces

A notion of disjointness in arbitrary partially ordered vector spaces is introduced by calling two elements x and y disjoint if the set of all upper bounds of x + y and −x − y equals the set of all upper bounds of x − y and −x + y. Several elementary properties are easily observed. The question whether the disjoint complement of a subset is a linear subspace appears to be more difficult. It is shown that in directed Archimedean spaces disjoint complements are always subspaces. The proof relies on theory on order dense embedding in vector lattices. In a non-Archimedean directed space even the disjoint complement of a singleton may fail to be a subspace. According notions of disjointness preserving operator, band, and band preserving operator are defined and some of their basic properties are studied.     

Covariance matrices and valuations

A complete classification of SL(n)$\mathrm{SL}(n)$ covariant matrix-valued valuations on functions with finite second moments is obtained. It is shown that there is a unique homogeneous such valuation. This valuation turns out to be the moment matrix.     

Positions of convex bodies associated to extremal problems and isotropic measures

We show that there are close relations between extremal problems in dual Brunn-Minkowski theory and isotropic-type properties for some Borel measures on the sphere. The methods we use allow us to obtain similar results in the context of Firey-Brunn-Minkowski theory. We also study reverse inequalities for dual mixed volumes which are related with classical positions, such as ?-position or isotropic position.     

Delta-semidefinite and Delta-convex Quadratic Forms in Banach Spaces

A continuous quadratic form (“quadratic form”, in short) on a Banach space X is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding symmetric linear operator factors through a Hilbert space; (b) delta-convex (i.e., representable as a difference of two continuous convex functions) if and only if T is a UMD-operator. It follows, for instance, that each quadratic form on an infinite-dimensional L _p (μ) space (1 ≤ p ≤ ∞) is: (a) delta-semidefinite iff p ≥ 2; (b) delta-convex iff p > 1. Some other related results concerning delta-convexity are proved and some open probms are stated. The first author was supported by NSF grant DMS-0555670. The second author was supported by the Russian Foundation for Basic Research, Grant 05-01-00066, and by Grant NSh-5813.2006.1. The third author was supported in part by the Ministero dell’Università e della Ricerca of Italy.     

Topological and bornological characterisations of ideals in von Neumann algebras: II

SupposeM is a von Neumann algebra on a Hilbert spaceH andI is any norm closed ideal inM. We extend to this setting the well known fact that the compact operators on a Hilbert space are precisely those whose restrictions to the closed unit ball are weak to norm continuous.     

Application of an idea of Vorono? to John type problems

Using an idea of Vorono?, many John type and minimum position problems in dimension d can be transformed into more accessible geometric problems on convex subsets of the -dimensional cone of positive definite quadratic forms. In this way, we prove several new John type and minimum position results and give alternative versions and extensions of known results. In particular, we characterize minimum ellipsoidal shells of convex bodies and, in the typical case, show their uniqueness and determine the contact number. These results are formulated also in terms of the circumradius of convex bodies. Next, circumscribed ellipsoids of minimum surface area of a convex body and the corresponding minimum position problem are studied. Then we investigate John type characterizations of minimum positions of a convex body with respect to moments and the product of a moment and the moment of the polar body. The technique used in this context, finally, is applied to obtain corresponding results for the mean width and the surface area.     

Ordered Involutive Operator Spaces

This is a companion to recent papers of the authors; here we construct the ‘noncommutative Shilov boundary’ of a (possibly nonunital) selfadjoint ordered space of Hilbert space operators. The morphisms in the universal property of the boundary preserve order. As an application, we consider ‘maximal’ and ‘minimal’ unitizations of such ordered operator spaces.     

Asymptotics of unitary and orthogonal matrix integrals

In this paper, we prove that in small parameter regions, arbitrary unitary matrix integrals converge in the large N limit and match their formal expansion. Secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the HCIZ integral. Our convergence result also leads us to new results of smoothness of microstates. We finally generalize our approach to integrals over the orthogonal group.     

Mixed Norm and Multidimensional Lorentz Spaces

In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved ([16], [7]). However, the question for multidimensional Lorentz spaces is still open. In this paper, we consider weights of product type, and give necessary and sufficient conditions for the Lorentz spaces, defined with respect to the two-dimensional decreasing rearrangement, to be normable. To this end, it is also useful to study the mixed norm Lorentz spaces. Finally, we prove embeddings between all the classical, multidimensional, and mixed norm Lorentz spaces. Research partially supported by KAW 2000.0048 and STINT KU 2002-4025. Research partially supported by Grants MTM2004-02299, 2005SGR00556 and The Swedish Research Council no. 624-2003-571.     

Epigraph of operator functions

It is known that a real function f is convex if and only if the set E(f) = {(x, y) ∈ ? × ?; f (x) ≤ y}, the epigraph of f is a convex set in ?². We state an extension of this result for operator convex functions and C^?-convex sets as well as operator log-convex functions and C^?-log-convex sets. Moreover, the C^?-convex hull of a Hermitian matrix has been represented in terms of its eigenvalues.     

Some Measurability Results and Applications to Spaces with Mixed Family-norm

A space with mixed family-norm consists of all functions x on a product space such that the function belongs to V (here, U(t) and V denote given Köthe spaces). Conditions for the measurability of y are given, and the Köthe dual of such spaces is determined. For this purpose a generalization of the Luxemburg-Gribanov theorem for ‘uniformly measurable’ functions is proved. This result is also formulated for vector functions.     

Asymptotic formulas for the diameter of sections of symmetric convex bodies

Sharpening work of the first two authors, for every proportion λ∈(0,1) we provide exact quantitative relations between global parameters of n-dimensional symmetric convex bodies and the diameter of their random ⌊λn⌋-dimensional sections. Using recent results of Gromov and Vershynin, we obtain an “asymptotic formula” for the diameter of random proportional sections.