Uniqueness of the topology on L1(G)

J. Extremera, J. F. Mena, A. R. Villena
2002 Studia Mathematica  
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 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
more » ... rs 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 < ∞, then L p (G) carries a unique Banach space topology that makes translations continuous.
doi:10.4064/sm150-2-5 fatcat:iumnhv6kwjakrdtdsmqkap4ire