Beurling–Figà-Talamanca–Herz algebras

Serap Öztop ¹   ; Volker Runde ²   ; Nico Spronk ³

¹ Department of Mathematics Faculty of Science Istanbul University Istanbul, Turkey
² Department of Mathematical and Statistical Sciences University of Alberta Edmonton, AB, Canada T6G 2G1
³ Department of Pure Mathematics University of Waterloo Waterloo, ON, Canada N2L 3G1

Studia Mathematica, Tome 210 (2012) no. 2, pp. 117-135

Résumé

For a locally compact group $G$ and $p \in (1,\infty)$, we define and study the Beurling–Figà-Talamanca–Herz algebras $A_p(G,\omega)$. For $p=2$ and abelian $G$, these are precisely the Beurling algebras on the dual group $\hat{G}$. For $p =2$ and compact $G$, our approach subsumes an earlier one by H. H. Lee and E. Samei. The key to our approach is not to define Beurling algebras through weights, i.e., possibly unbounded continuous functions, but rather through their inverses, which are bounded continuous functions. We prove that a locally compact group $G$ is amenable if and only if one—and, equivalently, every—Beurling–Figà-Talamanca–Herz algebra $A_p(G,\omega)$ has a bounded approximate identity.

Détail

DOI : 10.4064/sm210-2-2

Mots-clés : locally compact group infty define study beurling fig talamanca herz algebras omega abelian these precisely beurling algebras dual group hat compact approach subsumes earlier lee samei key approach define beurling algebras through weights possibly unbounded continuous functions rather through their inverses which bounded continuous functions prove locally compact group amenable only equivalently every beurling fig talamanca herz algebra omega has bounded approximate identity