Universal and homogeneous structures on the Urysohn and Gurarij spaces

Using Fraïssé theoretic methods we enrich the Urysohn universal space by universal and homogeneous closed relations, retractions, closed subsets of the product of the Urysohn space itself and some fixed compact metric space, L-Lipschitz map to a fixed Polish metric space. The latter lifts to a universal linear operator of norm L on the Lispchitz-free space of the Urysohn space.Moreover, we enrich the Gurarij space by a universal and homogeneous closed subspace and norm one projection onto a 1-complemented subspace. We construct the Gurarij space by the classical Fraïssé theoretic approach.     

Rethinking polyhedrality for lindenstrauss spaces

We present a Lindenstrauss space with an extreme point that does not contain a subspace linearly isometric to c. This example disproves a result stated by Zippin in a paper published in 1969 and it shows that some classical characterizations of polyhedral Lindenstrauss spaces, based on Zippin’s result, are false, whereas some others remain unproven; then we provide a correct proof for those characterizations. Finally, we also disprove a characterization of polyhedral Lindenstrauss spaces given by Lazar, in terms of the compact norm-preserving extension of compact operators, and we give an equivalent condition for a Banach space X to satisfy this property.     

A minimal 0–1 subshift with noncompact set of ergodic measures

Summary By a minimal 0–1 subshift we mean a pair (X, S), where S denotes the left shift on C={0, 1}^z and X is a minimal compact S-invariant subset of C. Developing some of the methods of Williams [2] of obtaining not uniquely ergodic minimal subshifts we construct such a subshift, for which the set of all ergodic measures is noncompact for the weak^* topology. In other words, the Choquet simplex of all invariant measures of the subshift is not a Bauer simplex.     

On separable Lindenstrauss spaces

Isometric embeddings from l_∞ⁿin l_∞^{n + 1} can be described by a_i,n, i ? n, with $\text{∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{¦ a}{}_{\mathrm{i,n}}\text{¦ ? 1}$, such that e_i,n = e_{i,n + 1} + a_i,ne_{n + 1,n + 1}; i = 1,…, n; holds, where e_i,nand e_{i,n + 1} are the elements of the canonical unit vector bases of l_∞ⁿand l_∞^{n + 1}, respectively (negative signs may occur). We study the connections between a triangular substochastic matrix A, whose nth column consists of the elements a_i,n, i = 1,…, n, and the Banach space a_i,n, E_n ? E_{n + 1}, E_n ? l_∞ⁿ, where A determines the embeddings of the E_n. The class of these Banach spaces is the class of all separable Lindenstrauss spaces. Sufficient and necessary conditions are stated for a matrix A to represent c₀and c. Furthermore, we characterize the class of all extreme triangular substochastic matrices which represents C(K), where K is the Cantor set. We investigate how the special biface structure of the dual unit ball of X is reflected in the elements of a matrix A representing the separable Lindenstrauss space X. This is applicable to Gurarij spaces; we give a new proof for the maximality property of Gurarij spaces and show that they are isomorphic to A(S) where S is a Choquet simplex with dense extreme points.     

The Choquet integral representability of comonotonically additive functionals in locally compact spaces

We give an alternative and direct approach to the Choquet integral representability of a comonotonically additive, bounded, monotone functional I defined on the space of all continuous, real-valued functions on a locally compact space X with compact support and on the space of all continuous, real-valued functions on X vanishing at infinity. To this end, we introduce the notion of the asymptotic translatability of the functional I and show that this simple notion is equivalent to the Choquet integral representability of I with respect to a monotone measure on X with appropriate regularity.     

The topological space of all extreme points of a compact convex set

G. Choquet and R. Haydon have proved that every Polish (i.e. completely metrizable and separable) spaceX is homeomorphic to the space of all extreme points of a metrizable compact convex setK which can be chosen to be even a Choquet simplex. In our paper we generalize partially this result to the case of a complete metric spaceX of zero covering dimension, without requiring any separability property forX.     

EveryL 1-predual is complemented in a simplex space

We show that everyL ₁-predual space is complemented in a simplex space. This answers a question raised by Lazar and Lindenstrauss.     

Vietoris topology on partial maps with compact domains

The space PK of partial maps with compact domains (identified with their graphs) forms a subspace of the hyperspace of nonempty compact subsets of a product space endowed with the Vietoris topology. Various completeness properties of PK, including ?ech-completeness, sieve completeness, strong Choquetness, and (hereditary) Baireness, are investigated. Some new results on the hyperspace K(X) of compact subsets of a Hausdorff X with the Vietoris topology are obtained; in particular, it is shown that there is a strongly Choquet X, with 1st category K(X).     

Conditions for Choquet integral representation of the comonotonically additive and monotone functional

If the universal set X is not compact but locally compact, a comonotonically additive and monotone functional (for short c.m.) on the class of continuous functions with compact support is not represented by one Choquet integral, but represented by the difference of two Choquet integrals. The conditions for which a c.m. functional can be represented by one Choquet integral are discussed.     

Choquet simplices as spaces of invariant probability measures on post-critical sets

A well-known consequence of the ergodic decomposition theorem is that the space of invariant probability measures of a topological dynamical system, endowed with the weak^∗ topology, is a non-empty metrizable Choquet simplex. We show that every non-empty metrizable Choquet simplex arises as the space of invariant probability measures on the post-critical set of a logistic map. Here, the post-critical set of a logistic map is the ω-limit set of its unique critical point. In fact we show the logistic map f can be taken in such a way that its post-critical set is a Cantor set where f   is minimal, and such that each invariant probability measure on this set has zero Lyapunov exponent, and is an equilibrium state for the potential −ln|f^′|$-\ln |f^{\prime }|$.     

A solution of the abstract Dirichlet problem for Baire-one functions

Let X be a compact convex subset of a locally convex space. We show that any bounded Baire-one function defined on can be extended to an affine Baire-one function on X if and only if X is a Choquet simplex and satisfies a certain topological property.     

Normality of products of monotonically normal spaces with compact spaces

Let S be the class of all spaces, each of which is homeomorphic to a stationary subset of a regular uncountable cardinal (depending on the space). In this paper, we prove the following result: The product X×C of a monotonically normal space X and a compact space C is normal if and only if S×C is normal for each closed subspace S in X belonging to S. As a corollary, we obtain the following result: If the product of a monotonically normal space and a compact space is orthocompact, then it is normal.     

On the ordered K0 group of a simple C*-algebra

It is shown that any countable simple and unperforated partially ordered Abelian group with order unit, paired in an arbitrary way with a compact metrizable Choquet simplex, occurs as the K₀group of a simple unital ^*-algebra which is the inductive limit of a sequence of subhomogeneous ^*-algebras of a particularly simple form.     

On complemented subspaces of simplex spaces

Jellett's theorem about the resolution of the space, A(X) of continuous affine functions on a compact ahoquet simplex X into a direct sum is generalized to simplex spaces. A new characteristic property of the space A(X), dual tol ¹(Γ), is given.     

Factoring compact operators

We consider the following problem: Does there exist a separable Banach spaceZ such that every compact operator can be factored as a productTS withT, S compact, rangeS=DomainT=Z? Our investigation yields a reasonable partial solution to this problem as well as the following independent result: A Banach space which has theλ-metric approximation property can be embedded as a complemented subspace of aπ _λ space.     

The Bishop-Phelps-Bollobás theorem for operators

We prove the Bishop-Phelps-Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop-Phelps-Bollobás theorem holds for operators from ?₁ into Y. Several examples of classes of such spaces are provided. For instance, the Bishop-Phelps-Bollobás theorem holds when the range space is finite-dimensional, an L₁(μ)-space for a σ-finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.     

The Choquet integral representation in the affine vector-valued case

LetX be any compact convex subset of a locally convex Hausdorff space andE be a complex Banach space. We denote byA(X, E) the space of all continuous and affineE-valued functions defined onX. In this paper we prove thatX is a Choquet simplex if and only if the dual ofA(X, E) is isometrically isomorphic by a selection map toM _m (X, E*), the space ofE*-valued,w*-regular boundary measures onX. This extends and strengthens a result of G. M. Ustinov. To do this we show that for any compact convex setX, each element of the dual ofA(X, E) can be represented by a measure inM _m (X, E*) with the same norm, and this representation is unique if and only ifX is a Choquet simplex. We also prove that ifX is metrizable andE is separable then there exists a selection map from the unit ball of the dual ofA(X, E) into the unit ball ofM _m (X, E*) which is weak* to weak*-Borel measurable.This work will constitute a portion of the author's Ph.D. Thesis at the University of Illinois.     

Operators on separable L1-predual spaces

We give an extension of the classical Bartle–Dunford–Schwartz theorem for weakly compact operators on a $C(K)$ space, to weakly compact operators on a separable $L^1$-predual space. Using this we show that for operators on these spaces, the set of weakly compact operators that attain their norm is dense in the space of weakly compact operators. For operators from the space of affine continuous functions on a metrizable Choquet simplex with values in an uniformly convex space, we show that the operator theoretic version of the Bishop–Phelps–Bollobás property is valid. This gives an extension of some recent work of Kim and Lee.     

Affine functions on simplexes and extreme operators

IfK a simplex andX a Banach space thenA(K, X) denotes the space of affine continuous functions fromK toX with the supremum norm. The extreme points of the closed unit ball ofA(K, X) are characterized,X being supposed to satisfy certain conditions. This characterization is used to investigate the extreme compact operators from a Banach spaceX to the spaceA(K)=A(K, (− ∞, ∞)). This note is part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Prof. A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank them for their helpful advice and kind encouragement.     

A note on the Brouwer dimension of chainable spaces

A general method produces from a compact Hausdorff space S a compact Hausdorff space T with IndT=IndS+1. We show that if S is chainable, then T is also chainable while DgT<IndT, where Dg denotes dimensionsgrad, the dimension in the original sense of Brouwer. This leads to a chainable, first countable, separable space Xn with DgXn<IndXn=n for each integer n>1.