Beurling–Figà-Talamanca–Herz algebras

Volume 210 / 2012

Serap Öztop, Volker Runde, Nico Spronk Studia Mathematica 210 (2012), 117-135 MSC: Primary 43A99; Secondary 22D12, 43A15, 43A32, 46H05, 46J10. DOI: 10.4064/sm210-2-2


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.


  • Serap ÖztopDepartment of Mathematics
    Faculty of Science
    Istanbul University
    Istanbul, Turkey
  • Volker RundeDepartment of Mathematical
    and Statistical Sciences
    University of Alberta
    Edmonton, AB, Canada T6G 2G1
  • Nico SpronkDepartment of Pure Mathematics
    University of Waterloo
    Waterloo, ON, Canada N2L 3G1

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