Banach spaces of continuous functions with few operators

We present two constructions of infinite, separable, compact Hausdorff spaces K for which the Banach space C(K) of all continuous real-valued functions with the supremum norm has remarkable properties. In the first construction K is zero-dimensional and C(K) is non-isomorphic to any of its proper subspaces nor any of its proper quotients. In particular, it is an example of a C(K) space where the hyperplanes, one co-dimensional subspaces of C(K), are not isomorphic to C(K). In the second construction K is connected and C(K) is indecomposable which implies that it is not isomorphic to any C(K) for K zero-dimensional. All these properties follow from the fact that there are few operators on our C(K)s. If we assume the continuum hypothesis the spaces have few operators in the sense that every linear bounded operator T : C (K) C (K) is of the form gI+S where gC(K) and S is weakly compact or equivalently (in C(K) spaces) strictly singular.While conducting research leading to the results presented in this paper, the author was partially supported by a fellowship Produtividade em Pesquisa from National Research Council of Brazil (Conselho Nacional de Pesquisa, Processo Número 300369/01-8). The final stage of the research was realized at the Fields Institute in Toronto where the author was supported by the State of São Paulo Research Assistance Foundation (Fundação de Amparoá Pesquisa do Estado de São Paulo), Processo Número 02/03677-7 and by the Fields Institute.Revised version: 29 January 2004     

Abstract functions with continuous differences and Namioka spaces

Let be a semigroup and a topological space. Let be an Abelian topological group. The right differences of a function are defined by for . Let be continuous at the identity of for all in a neighbourhood of . We give conditions on or range under which is continuous for any topological space . We also seek conditions on under which we conclude that is continuous at for arbitrary . This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.

     

Shadowing with multidimensional time in Banach spaces

We study linear dynamical systems with multidimensional time in Banach spaces. Using Taylor functional calculus we prove that under additional assumptions hyperbolic systems have shadowing property.     

Separately continuous mappings with values in nonlocally convex spaces

We prove that a collection (X, Y, Z) is a Lebesgue triple if X is a metrizable space, Y is a perfectly normal space, and Z is a strongly σ-metrizable topological vector space with stratification (Z _m) _{m = 1}^∞, where, for every m ∈ ℕ, Z _m is a closed, metrizable, separable subspace of Z that is arcwise connected and locally arcwise connected. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1639–1646, December, 2007.     

Function spaces from core compact coherent spaces to continuous B-domains

It is proved in this paper that for a continuous B-domain L, the function space [X→L] is continuous for each core compact and coherent space X. Further, applications are given. It is proved that:

(1)

the function space from the unit interval to any bifinite domain which is not an L-domain is not Lawson compact;

(2)

the Isbell and Scott topologies on [X→L] agree for each continuous B-domain L and core compact coherent space X.

     

Extensions by spaces of continuous functions

We present two complementary results on the splitting of exact sequences having the form . The first one characterizes the Banach spaces such that for every compact space . The second is a nonlinear generalization of Zippin's criterion for the extension of -valued operators.

     

Weakly continuous mappings on Banach spaces

It is shown that every n-homogeneous continuous polynomial on a Banach space E which is weakly continuous on the unit ball of E is weakly uniformly continuous on the unit ball of E. Applications of the result to spaces of polynomials and holomorphic mappings on E are given.     

On isomorphisms of continuous function spaces

The following theorem, which strengthens the classical theorem of Stone, is proved: If there is an isomorphism φ ofC(X) ontoC(Y) (X, Y-compact) with ∥ φ ∥ · ∥ φ^?1 ∥ < 2, thenX andY are homeomorphic.     

Dense-lineability in spaces of continuous functions

In this paper we provide a general method to prove that certain nonlinear families of continuous functions contain dense linear manifolds. An application is furnished.

     

Weighted spaces of vector-valued continuous functions

Summary We study completeness of weighted spaces of vector-valued functions. Also, the dual of a weighted space of vector-valued continuous functions on a locally compact space is characterized as a space of vector valued Radon measures with values in the topological dual of the range space. Research supported by a grant from the National Science Foundation, GP 22713. Entrata in Redazione il 25 ottobre 1970.     

Riesz spaces of real continuous functions

An Archimedean Riesz space E is isomorphic to C(X) for some completely regular Hausdorff space X if and only if there exists a weak order unit e > 0 for which E is e-uniformly complete, e-semisimple, e-separating and 2-universally e-complete.     

Characterizing continuous functions on compact spaces

We consider the following problem: given a set X and a function , does there exist a compact Hausdorff topology on X which makes T continuous? We characterize such functions in terms of their orbit structure. Given the generality of the problem, the characterization turns out to be surprisingly simple and elegant. Amongst other results, we also characterize homeomorphisms on compact metric spaces.     

Distance to spaces of continuous functions

We link here distances between iterated limits, oscillations, and distances to spaces of continuous functions. For a compact space K, a uniformly bounded set H of the space of real-valued continuous functions C(K), and ε?0, we say that H ε-interchanges limits with K, if the inequality

     

Free continuous actions on zero-dimensional spaces

We show that every countably infinite group admits a free, continuous action on the Cantor set having an invariant probability measure. We also show that every countably infinite group admits a free, continuous action on a non-homogeneous compact metric space and the action is minimal (that is to say, every orbit is dense). In answer to a question posed by Giordano, Putnam and Skau, we establish that there is a continuous, minimal action of a countably infinite group on the Cantor set such that no free continuous action of any group gives rise to the same equivalence relation.     

Smooth spaces versus continuous spaces in models for synthetic differential geometry

In topos models for synthetic differential geometry we study connections between smooth spaces (which interpret synthetic calculus) and continuous spaces (which interpret intuitionistic analysis). Our main tools are adjoint retractions of toposes and the standard map from the smooth reals to the continuous reals.     

Universal spaces for the subdifferentials of sublinear operators with values in the spaces of continuous functions

Given a continuous sublinear operator P: V → C(X) from a Hausdorff separable locally convex space V to the Banach space C(X) of continuous functions on a compact set X we prove that the subdifferential ∂P at zero is operator-affinely homeomorphic to the compact subdifferential ∂ ^c Q, i.e., the subdifferential consisting only of compact linear operators, of some compact sublinear operator Q: ł² → C(X) from a separable Hilbert space ł², where the spaces of operators are endowed with the pointwise convergence topology. From the topological viewpoint, this means that the space L ^c (ł², C(X)) of compact linear operators with the pointwise convergence topology is universal with respect to the embedding of the subdifferentials of sublinear operators of the class under consideration.