Uniqueness of the topology on $L^1(G)$

J. Extremera¹ ; J. F. Mena² ; A. R. Villena²

¹Departamento de Análisis Matemático Facultad de Ciencias Universidad de Granada 18071 Granada, Spain
²Departamento de Análisis Matemático Facultad de Ciencias Universidad de Granada 18071 GRANADA, Spain

Studia Mathematica, Tome 150 (2002) no. 2, pp. 163-173

Résumé

Let $G$ be a locally compact abelian group and let $X$ be a translation invariant linear subspace of $L^1(G)$. If $G$ is noncompact, then there is at most one Banach space topology on $X$ that makes translations on $X$ continuous. In fact, the Banach space topology on $X$ is determined just by a single nontrivial translation in the case where the dual group $ \widehat {G} $ is connected. For $G$ compact we show that the problem of determining a Banach space topology on $X$ by considering translation operators on $X$ is closely related to the classical problem of determining whether or not there is a discontinuous translation invariant linear functional on $X$. As a matter of fact $L^1(G)$ does not carry a unique Banach space topology that makes translations continuous, but translations almost determine the Banach space topology on $X$. Moreover, if $G$ is connected and compact and $1 p \infty $, then $L^p(G)$ carries a unique Banach space topology that makes translations continuous.

Détail

DOI : 10.4064/sm150-2-5

Keywords: locally compact abelian group translation invariant linear subspace noncompact there banach space topology makes translations continuous banach space topology determined just single nontrivial translation where dual group widehat connected compact problem determining banach space topology considering translation operators closely related classical problem determining whether there discontinuous translation invariant linear functional matter does carry unique banach space topology makes translations continuous translations almost determine banach space topology moreover connected compact infty carries unique banach space topology makes translations continuous