The order structure of the space of measures with continuous translation
Annales de l'Institut Fourier, Volume 32 (1982) no. 2, p. 67-110

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.

Soit G un groupe localement compact, et soit B une norme de fonctions (c’est-à-dire ayant la propriété de Riesz) sur L 1 (G) loc , telle que le sous-espace L (G,B), formé des fonctions localement intégrables de B -norme bornée, soit un espace de fonctions de Banach invariant et solide (solide dans l’espace de Riesz L 1 (G) loc ). Considérons l’espace L RUC (G,B), formé des fonctions dans L (G,B) avec une translation à droite qui est une application continue de G dans L (G,B). On trouvera les caractérisations du cas où L RUC (G,B) est un sous-espace solide (un idéal de Riesz). Ces descriptions sont données à l’aide de la continuité pour l’ordre de la norme B sur certains sous-espaces de L (G). La discussion entière se déroule et les résultats sont formulés dans le contexte des semi-groupes fondamentaux ayant un élément neutre. Tout groupe localement compact est un cas spécial d’un tel semi-groupe.

@article{AIF_1982__32_2_67_0,
     author = {Sleijpen, G\'erard L. G.},
     title = {The order structure of the space of measures with continuous translation},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {32},
     number = {2},
     year = {1982},
     pages = {67-110},
     doi = {10.5802/aif.873},
     zbl = {0468.43001},
     mrnumber = {83k:43005},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1982__32_2_67_0}
}
Sleijpen, Gérard L. G. The order structure of the space of measures with continuous translation. Annales de l'Institut Fourier, Volume 32 (1982) no. 2, pp. 67-110. doi : 10.5802/aif.873. http://www.numdam.org/item/AIF_1982__32_2_67_0/

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