Uniqueness of the topology on L¹(G)

• Autorzy: J. Extremera , J. F. Mena , A. R. Villena
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 43A15: L p -spaces and other function spaces on groups, semigroups, etc.
• 46H40: Automatic continuity
• 43A20: L 1 -algebras on groups, semigroups, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2002
• Tom: 150
• Numer: 2
• Strony: 163-173

Daty

wydano

2002

Twórcy

autor

J. Extremera

• Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

autor

J. F. Mena

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

autor

A. R. Villena

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

Identyfikatory

DOI

10.4064/sm150-2-5