The order structure of the space of measures with continuous translation

Sleijpen, Gérard L. G.

Annales de l'Institut Fourier, Tome 32 (1982) no. 2, pp. 67-110.

Soit $G$ un groupe localement compact, et soit $\parallel {\parallel }_{\infty }^B$ une norme de fonctions (c’est-à-dire ayant la propriété de Riesz) sur $L^1(G{)}_{\mathrm{loc}}$, telle que le sous-espace $L^{\infty }(G,B)$, formé des fonctions localement intégrables de $\parallel {\parallel }_{\infty }^B$-norme bornée, soit un espace de fonctions de Banach invariant et solide (solide dans l’espace de Riesz $L^1(G{)}_{\mathrm{loc}}$). Considérons l’espace $L_{\mathrm{RUC}}(G,B)$, 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 $L_{\mathrm{RUC}}(G,B)$ 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 $\parallel {\parallel }_{\infty }^B$ sur certains sous-espaces de $L^{\infty }\left(G\right)$. 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 $\parallel {\parallel }_{\infty }^B$ be a function norm on $L^1(G{)}_{\mathrm{loc}}$ such that the space $L^{\infty }(G,B)$ of all locally integrable functions with finite $\parallel {\parallel }_{\infty }^B$-norm is an invariant solid Banach function space. Consider the space $L_{\mathrm{RUC}}(G,B)$ 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 $L_{\mathrm{RUC}}(G,B)$ is a Riesz ideal of $L^{\infty }(G,B)$ are given in terms of the order-continuity of $\parallel {\parallel }_{\infty }^B$ on certain subspaces of $L^{\infty }\left(G\right)$. 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.

[1] A. C. Baker and J. W. Baker, Algebras of measures on a locally compact semigroup II, J. London Math. Soc. (2), 4 (1972), 685-695. | MR | Zbl

[2] J.-P. Bertrandias, Unions et intersections d'espaces Lp sur un espace localement compact, Bull. Sc. Math., (2), 101 (1977), 209-247. | MR | Zbl

[3] H. A. M. Dzinotyiweyi and G. L. G. Sleijpen, A note on measures on foundation semigroups with weakly compact orbits, Pac. Journal Math. (1), 81 (1979), 61-69. | MR | Zbl

[4] D. A. Edwards, On translates of L∞-functions, J. London Math. Soc., 36 (1961), 431-432. | MR | Zbl

[5] R. E. Edwards, E. Hewitt and G. Ritter, Fourier multipliers for certain spaces of functions with compact support, Inventiones Math., 40 (1977), 37-57. | EuDML | MR | Zbl

[6] H. G. Feichtinger, On a class of convolution algebras of functions, Ann. Inst. Fourier, Grenoble, 27, 3 (1977), 135-162. | EuDML | Numdam | MR | Zbl

[7] H. G. Feichtinger, Multipliers of Banach spaces of functions on groups, Math. Z., 152 (1976), 47-58. | EuDML | MR | Zbl

[8] H. G. Feichtinger, A characterization of Wiener's algebra on locally compact groups, Arch. Math., 39 (1977), 136-140. | MR | Zbl

[9] D. H. Fremlin, Topological Riesz-spaces and measure theory, Cambridge Univ. Press, (1974). | MR | Zbl

[10] G. G. Gould, On a class of integration spaces, J. London Math. Soc., 34 (1959), 161-172. | MR | Zbl

[11] H. Kharaghani, The weaky continuous left translations of measures with applications to invariant means, preprint.

[12] T.-S. Liu, A. C. M. Van Rooij and J.-K. Wang, On some group modules related to Wiener's algebra M1, Pac. Journal Math. (2), 55 (1974), 507-520. | Zbl

[13] H. Porta, L. A. Rubel and A. L. Shields, Separability of orbits of functions on locally compact groups, Studia Math., 48 (1973), 89-94. | MR | Zbl

[14] H. H. Schaeffer, Banach-Lattices and Positive Operators, Springer-Verlag, Berlin Heidelberg New-York, (1974). | Zbl

[15] G. L. G. Sleijpen, Locally compact semigroups and continuous translations of measures, Proc. London Math. Soc., (3), 37 (1978), 75-97. | MR | Zbl

[16] G. L. G. Sleijpen, Emaciated sets and measures with continuous translations, Proc. London Math. Soc., (3), 37 (1978), 98-119. | MR | Zbl

[17] G. L. G. Sleijpen, L-multipliers for foundation semigroups with identity element, Proc. London Math. Soc., (3), 39 (1979), 299-330. | MR | Zbl

[18] G. L. G. Sleijpen, The support of the Wiener algebra on stips, Indag. Math., 42 (1980), 61-82. | MR | Zbl

[19] G. L. G. Sleijpen, Lp-spaces on foundation semigroups with identity element, Report 7906, Mathematical Institute, Catholic University, Nijmegen (1979).

[20] G. L. G. Sleijpen, Convolution measure algebras on semigroups, Thesis, Catholic University, Nijmegen (1976).