Linear continuous surjections of Cp-spaces over compacta

Kazuhiro Kawamura; Arkady Leiderman

doi:10.1016/j.topol.2017.01.022

Linear continuous surjections of C_p-spaces over compacta

Kazuhiro Kawamura, Arkady Leiderman

Department of Mathematics

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

7 Scopus citations

Abstract

Let X and Y be compact Hausdorff spaces and suppose that there exists a linear continuous surjection T:C_p(X)→C_p(Y), where C_p(X) denotes the space of all real-valued continuous functions on X endowed with the pointwise convergence topology. We prove that dim⁡X=0 implies dim⁡Y=0. This generalizes a previous theorem [7, Theorem 3.4] for compact metrizable spaces. Also we point out that the function space C_p(P) over the pseudo-arc P admits no densely defined linear continuous operator C_p(P)→C_p([0,1]) with a dense image.

Original language	English
Pages (from-to)	135-145
Number of pages	11
Journal	Topology and its Applications
Volume	227
DOIs	https://doi.org/10.1016/j.topol.2017.01.022
State	Published - 15 Aug 2017

Keywords

C-theory
Dimension
Hereditarily indecomposable continua
Linear operators

ASJC Scopus subject areas

Geometry and Topology

Access to Document

10.1016/j.topol.2017.01.022

Cite this

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title = "Linear continuous surjections of Cp-spaces over compacta",

abstract = "Let X and Y be compact Hausdorff spaces and suppose that there exists a linear continuous surjection T:Cp(X)→Cp(Y), where Cp(X) denotes the space of all real-valued continuous functions on X endowed with the pointwise convergence topology. We prove that dim⁡X=0 implies dim⁡Y=0. This generalizes a previous theorem [7, Theorem 3.4] for compact metrizable spaces. Also we point out that the function space Cp(P) over the pseudo-arc P admits no densely defined linear continuous operator Cp(P)→Cp([0,1]) with a dense image.",

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AU - Leiderman, Arkady

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N2 - Let X and Y be compact Hausdorff spaces and suppose that there exists a linear continuous surjection T:Cp(X)→Cp(Y), where Cp(X) denotes the space of all real-valued continuous functions on X endowed with the pointwise convergence topology. We prove that dim⁡X=0 implies dim⁡Y=0. This generalizes a previous theorem [7, Theorem 3.4] for compact metrizable spaces. Also we point out that the function space Cp(P) over the pseudo-arc P admits no densely defined linear continuous operator Cp(P)→Cp([0,1]) with a dense image.

AB - Let X and Y be compact Hausdorff spaces and suppose that there exists a linear continuous surjection T:Cp(X)→Cp(Y), where Cp(X) denotes the space of all real-valued continuous functions on X endowed with the pointwise convergence topology. We prove that dim⁡X=0 implies dim⁡Y=0. This generalizes a previous theorem [7, Theorem 3.4] for compact metrizable spaces. Also we point out that the function space Cp(P) over the pseudo-arc P admits no densely defined linear continuous operator Cp(P)→Cp([0,1]) with a dense image.

KW - C-theory

KW - Dimension

KW - Hereditarily indecomposable continua

KW - Linear operators

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