Operators determining the complete norm topology of C(K)

A. Villena

Studia Mathematica (1997)

  • Volume: 124, Issue: 2, page 155-160
  • ISSN: 0039-3223

Abstract

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For any uniformly closed subalgebra A of C(K) for a compact Hausdorff space K without isolated points and x 0 ∈ A , we show that every complete norm on A which makes continuous the multiplication by x 0 is equivalent to ∥ · ∥ ∞ provided that x 0 - 1 ( λ ) has no interior points whenever λ lies in ℂ. Actually, these assertions are equivalent if A = C(K).

How to cite

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Villena, A.. "Operators determining the complete norm topology of C(K)." Studia Mathematica 124.2 (1997): 155-160. <http://eudml.org/doc/216404>.

@article{Villena1997,
abstract = {For any uniformly closed subalgebra A of C(K) for a compact Hausdorff space K without isolated points and $x_\{0\} ∈ A$, we show that every complete norm on A which makes continuous the multiplication by $x_\{0\}$ is equivalent to $∥·∥_\{∞\}$ provided that $x_\{0\}^\{-1\}(λ)$ has no interior points whenever λ lies in ℂ. Actually, these assertions are equivalent if A = C(K).},
author = {Villena, A.},
journal = {Studia Mathematica},
keywords = {uniformly closed subalgebra; complete norm},
language = {eng},
number = {2},
pages = {155-160},
title = {Operators determining the complete norm topology of C(K)},
url = {http://eudml.org/doc/216404},
volume = {124},
year = {1997},
}

TY - JOUR
AU - Villena, A.
TI - Operators determining the complete norm topology of C(K)
JO - Studia Mathematica
PY - 1997
VL - 124
IS - 2
SP - 155
EP - 160
AB - For any uniformly closed subalgebra A of C(K) for a compact Hausdorff space K without isolated points and $x_{0} ∈ A$, we show that every complete norm on A which makes continuous the multiplication by $x_{0}$ is equivalent to $∥·∥_{∞}$ provided that $x_{0}^{-1}(λ)$ has no interior points whenever λ lies in ℂ. Actually, these assertions are equivalent if A = C(K).
LA - eng
KW - uniformly closed subalgebra; complete norm
UR - http://eudml.org/doc/216404
ER -

References

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  1. [1] B. E. Johnson, The uniqueness of the (complete) norm topology, Bull. Amer. Math. Soc. 73 (1967), 537-539. Zbl0172.41004
  2. [2] A. Rodríguez, The uniqueness of the complete norm topology in complete normed nonassociative algebras, J. Funct. Anal. 60 (1985), 1-15. Zbl0602.46055
  3. [3] Z. Semadeni, Banach Spaces of Continuous Functions, I, Polish Sci. Publ., 1971. 
  4. [4] A. M. Sinclair, Automatic Continuity of Linear Operators, Cambridge University Press, 1976. Zbl0313.47029

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