Uniqueness of the topology on $L^1(G)$
Volume 150 / 2002
Abstract
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.
