The order structure of the space of measures with continuous translation

Gérard L. G. Sleijpen

Annales de l'institut Fourier (1982)

  • Volume: 32, Issue: 2, page 67-110
  • ISSN: 0373-0956

Abstract

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Let G be a locally compact group, and let ∥ ∥ ∞ B be a function norm on L 1 ( G ) loc such that the space L ∞ ( G , B ) of all locally integrable functions with finite ∥ ∥ ∞ B -norm is an invariant solid Banach function space. Consider the space L RUC ( G , B ) of all functions in L ∞ ( G , B ) of which the right translation is a continuous map from G into L ∞ ( G , B ) . Characterizations of the case where L RUC ( G , B ) is a Riesz ideal of L ∞ ( G , B ) are given in terms of the order-continuity of ∥ ∥ ∞ B on certain subspaces of L ∞ ( G ) . Throughout the paper, the discussion is carried out in the context of and all the results are formulated for foundation semigroups with identity element; any locally compact group is an example of such a semigroup.

How to cite

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Sleijpen, Gérard L. G.. "The order structure of the space of measures with continuous translation." Annales de l'institut Fourier 32.2 (1982): 67-110. <http://eudml.org/doc/74542>.

@article{Sleijpen1982,
abstract = {Let $G$ be a locally compact group, and let $\Vert ~~\Vert ^B_\infty $ be a function norm on $L^1(G)_\{\rm loc\}$ such that the space $L^\infty (G,B)$ of all locally integrable functions with finite $\Vert ~~\Vert ^B_\infty $-norm is an invariant solid Banach function space. Consider the space $L_\{\rm RUC\}(G,B)$ of all functions in $L^\infty (G,B)$ of which the right translation is a continuous map from $G$ into $L^\infty (G,B)$. Characterizations of the case where $L_\{\rm RUC\}(G,B)$ is a Riesz ideal of $L^\infty (G,B)$ are given in terms of the order-continuity of $\Vert ~~\Vert ^B_\infty $ on certain subspaces of $L^\infty (G)$. Throughout the paper, the discussion is carried out in the context of and all the results are formulated for foundation semigroups with identity element; any locally compact group is an example of such a semigroup.},
author = {Sleijpen, Gérard L. G.},
journal = {Annales de l'institut Fourier},
keywords = {Riesz ideal; function spaces with continuous translation},
language = {eng},
number = {2},
pages = {67-110},
publisher = {Association des Annales de l'Institut Fourier},
title = {The order structure of the space of measures with continuous translation},
url = {http://eudml.org/doc/74542},
volume = {32},
year = {1982},
}

TY - JOUR
AU - Sleijpen, Gérard L. G.
TI - The order structure of the space of measures with continuous translation
JO - Annales de l'institut Fourier
PY - 1982
PB - Association des Annales de l'Institut Fourier
VL - 32
IS - 2
SP - 67
EP - 110
AB - Let $G$ be a locally compact group, and let $\Vert ~~\Vert ^B_\infty $ be a function norm on $L^1(G)_{\rm loc}$ such that the space $L^\infty (G,B)$ of all locally integrable functions with finite $\Vert ~~\Vert ^B_\infty $-norm is an invariant solid Banach function space. Consider the space $L_{\rm RUC}(G,B)$ of all functions in $L^\infty (G,B)$ of which the right translation is a continuous map from $G$ into $L^\infty (G,B)$. Characterizations of the case where $L_{\rm RUC}(G,B)$ is a Riesz ideal of $L^\infty (G,B)$ are given in terms of the order-continuity of $\Vert ~~\Vert ^B_\infty $ on certain subspaces of $L^\infty (G)$. Throughout the paper, the discussion is carried out in the context of and all the results are formulated for foundation semigroups with identity element; any locally compact group is an example of such a semigroup.
LA - eng
KW - Riesz ideal; function spaces with continuous translation
UR - http://eudml.org/doc/74542
ER -

References

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