The order structure of the space of measures with continuous translation

Sleijpen, Gérard L. G.

Annales de l'Institut Fourier, Tome 32 (1982), p. 67-110 / Harvested from Numdam

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Abstract

Soit $G$ un groupe localement compact, et soit $∥ ∥_{\infty}^{B}$ une norme de fonctions (c’est-à-dire ayant la propriété de Riesz) sur $L^{1} (G)_{loc}$ , telle que le sous-espace $L^{\infty} (G, B)$ , formé des fonctions localement intégrables de $∥ ∥_{\infty}^{B}$ -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 $L^{\infty} (G, B)$ avec une translation à droite qui est une application continue de $G$ dans $L^{\infty} (G, B)$ . 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 $∥ ∥_{\infty}^{B}$ sur certains sous-espaces de $L^{\infty} (G)$ . 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.

Let $G$ be a locally compact group, and let $∥ ∥_{\infty}^{B}$ be a function norm on $L^{1} (G)_{loc}$ such that the space $L^{\infty} (G, B)$ of all locally integrable functions with finite $∥ ∥_{\infty}^{B}$ -norm is an invariant solid Banach function space. Consider the space $L_{RUC} (G, B)$ 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 $L_{RUC} (G, B)$ is a Riesz ideal of $L^{\infty} (G, B)$ are given in terms of the order-continuity of $∥ ∥_{\infty}^{B}$ 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.

Publié le : 1982-01-01

MR 83k:43005 | Zbl 0468.43001

DOI : https://doi.org/10.5802/aif.873

@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},
     volume = {32},
     year = {1982},
     pages = {67-110},
     doi = {10.5802/aif.873},
     mrnumber = {83k:43005},
     zbl = {0468.43001},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 32 (1982) pp. 67-110. doi : 10.5802/aif.873. http://gdmltest.u-ga.fr/item/AIF_1982__32_2_67_0/

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