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

top
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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.