Let G be a locally compact abelian group and let X be a translation invariant linear subspace of L¹(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 Ĝ 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 L¹(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 < ∞, then carries a unique Banach space topology that makes translations continuous.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm150-2-5,
author = {J. Extremera and J. F. Mena and A. R. Villena},
title = {Uniqueness of the topology on L$^1$(G)},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {163-173},
zbl = {0995.43002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm150-2-5}
}
J. Extremera; J. F. Mena; A. R. Villena. Uniqueness of the topology on L¹(G). Studia Mathematica, Tome 151 (2002) pp. 163-173. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm150-2-5/