Uniqueness of the topology on L¹(G)

J. Extremera, J. F. Mena, and A. R. Villena. "Uniqueness of the topology on L¹(G)." Studia Mathematica 150.2 (2002): 163-173. <http://eudml.org/doc/285222>.

@article{J2002,
abstract = {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 $L^\{p\}(G)$ carries a unique Banach space topology that makes translations continuous.},
author = {J. Extremera, J. F. Mena, A. R. Villena},
journal = {Studia Mathematica},
keywords = {locally compact Abelian group; Banach space topology; translation operator; compact Abelian group},
language = {eng},
number = {2},
pages = {163-173},
title = {Uniqueness of the topology on L¹(G)},
url = {http://eudml.org/doc/285222},
volume = {150},
year = {2002},
}

TY - JOUR
AU - J. Extremera
AU - J. F. Mena
AU - A. R. Villena
TI - Uniqueness of the topology on L¹(G)
JO - Studia Mathematica
PY - 2002
VL - 150
IS - 2
SP - 163
EP - 173
AB - 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 $L^{p}(G)$ carries a unique Banach space topology that makes translations continuous.
LA - eng
KW - locally compact Abelian group; Banach space topology; translation operator; compact Abelian group
UR - http://eudml.org/doc/285222
ER -