A note on Banach spaces E for which Ew is homeomorphic to Cp(X)

Jerzy Ka̧kol; Arkady Leiderman; Artur Michalak

doi:https://doi.org/10.1007/s13398-022-01292-8

A note on Banach spaces E for which E_w is homeomorphic to C_p(X)

Jerzy Ka̧kol, Arkady Leiderman, Artur Michalak

Department of Mathematics

Ben-Gurion University of the Negev

Research output: Contribution to journal › Article › peer-review

Abstract

C_p(X) denotes the space of continuous real-valued functions on a Tychonoff space X endowed with the topology of pointwise convergence. A Banach space E equipped with the weak topology is denoted by E_w. It is unknown whether C_p(K) and C(L) _w can be homeomorphic for infinite compact spaces K and L (Krupski, Rev R Acad Cienc Exact Fis Nat Ser A Mat (RACSAM) 110:557–563, 2016; Krupski and Marciszewski, J Math Anal Appl 452:646–658, 2017). In this paper we deal with a more general question: does there exist a Banach space E such that E_w is homeomorphic to the space C_p(X) for some infinite Tychonoff space X? We show that if such homeomorphism exists, then (a) X is a countable union of compact sets X_n, n∈ ω, where at least one component X_n is non-scattered; (b) the Banach space E necessarily contains an isomorphic copy of the Banach space ℓ₁.

Original language	American English
Article number	150
Journal	Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Volume	116
Issue number	4
DOIs	https://doi.org/10.1007/s13398-022-01292-8
State	Published - 1 Oct 2022

Keywords

Banach space
C(X) space
Homeomorphism
Weak topology

All Science Journal Classification (ASJC) codes

Analysis
Algebra and Number Theory
Geometry and Topology
Computational Mathematics
Applied Mathematics

Access to Document

https://doi.org/10.1007/s13398-022-01292-8

Cite this

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title = "A note on Banach spaces E for which Ew is homeomorphic to Cp(X)",

abstract = "Cp(X) denotes the space of continuous real-valued functions on a Tychonoff space X endowed with the topology of pointwise convergence. A Banach space E equipped with the weak topology is denoted by Ew. It is unknown whether Cp(K) and C(L) w can be homeomorphic for infinite compact spaces K and L (Krupski, Rev R Acad Cienc Exact Fis Nat Ser A Mat (RACSAM) 110:557–563, 2016; Krupski and Marciszewski, J Math Anal Appl 452:646–658, 2017). In this paper we deal with a more general question: does there exist a Banach space E such that Ew is homeomorphic to the space Cp(X) for some infinite Tychonoff space X? We show that if such homeomorphism exists, then (a) X is a countable union of compact sets Xn, n∈ ω, where at least one component Xn is non-scattered; (b) the Banach space E necessarily contains an isomorphic copy of the Banach space ℓ1.",

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author = "Jerzy K{\c a}kol and Arkady Leiderman and Artur Michalak",

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AU - Michalak, Artur

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N2 - Cp(X) denotes the space of continuous real-valued functions on a Tychonoff space X endowed with the topology of pointwise convergence. A Banach space E equipped with the weak topology is denoted by Ew. It is unknown whether Cp(K) and C(L) w can be homeomorphic for infinite compact spaces K and L (Krupski, Rev R Acad Cienc Exact Fis Nat Ser A Mat (RACSAM) 110:557–563, 2016; Krupski and Marciszewski, J Math Anal Appl 452:646–658, 2017). In this paper we deal with a more general question: does there exist a Banach space E such that Ew is homeomorphic to the space Cp(X) for some infinite Tychonoff space X? We show that if such homeomorphism exists, then (a) X is a countable union of compact sets Xn, n∈ ω, where at least one component Xn is non-scattered; (b) the Banach space E necessarily contains an isomorphic copy of the Banach space ℓ1.

AB - Cp(X) denotes the space of continuous real-valued functions on a Tychonoff space X endowed with the topology of pointwise convergence. A Banach space E equipped with the weak topology is denoted by Ew. It is unknown whether Cp(K) and C(L) w can be homeomorphic for infinite compact spaces K and L (Krupski, Rev R Acad Cienc Exact Fis Nat Ser A Mat (RACSAM) 110:557–563, 2016; Krupski and Marciszewski, J Math Anal Appl 452:646–658, 2017). In this paper we deal with a more general question: does there exist a Banach space E such that Ew is homeomorphic to the space Cp(X) for some infinite Tychonoff space X? We show that if such homeomorphism exists, then (a) X is a countable union of compact sets Xn, n∈ ω, where at least one component Xn is non-scattered; (b) the Banach space E necessarily contains an isomorphic copy of the Banach space ℓ1.

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