Separating maps between Lebesgue-Fourier algebras

Details Separating maps between Lebesgue-Fourier algebras Separating maps between Lebesgue-Fourier algebras

2011-11-25

arxiv.org/abs/1111.5922v1

Mathematics

Functional Analysis

Scientific paper

Let $G_{1}$ and $G_{2}$ be locally compact groups; it is known that ${\cal L}A(G_{i})=L^1(G_i)\cap A(G_i)$ is an abstract Segal algebra with respect to $A(G_{i})$ for $i = 1,2$. A linear mapping $T:{\cal L}A(G_{1}) \rightarrow {\cal L}A(G_{2})$ is a {\it separating} map if $f\cdot g \equiv0$ implies $Tf \cdot Tg \equiv 0$ for $f,g \in {\cal L}A(G_{1})$. In this paper, we show that a separating bijective map is always continuous and also, that there exist some extensions of $T$ to the larger algebras. We introduce a certain condition (condition $(P)$) under which the existence of a bijective separating map leads to existence of a topological isomorphism between $G_{1}$ and $G_2$. We also characterize bijective separating maps as a weighted isomorphism on locally compact amenable groups. Moreover, we derive some similar results for Lebesgue-Fourier algebras considered as Segal algebras for locally compact abelian groups.

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