Beurling-Figà-Talamanca-Herz algebras (Q2907265)

Beurling algebras are weighted group algebras \(L^1(G,\omega)\) on locally compact groups \(G\). The Fourier algebra \(A(G)\) was defined and studied in the pioneering work of P. Eymard. For abelian \(G\), \(A(G)\) is identified with the group algebra \(L^1(\widehat G)\) on the dual group. Naturally, people have tried to look for `weighted' Fourier algebras or Beurling-Fourier algebras on nonabelian groups -- \textit{J. Ludwig} et al. [J. Funct. Anal. 262, No. 2, 463--499 (2012; Zbl 1235.43003)] and \textit{H. H Lee} and \textit{E. Samei} [J. Funct. Anal. 262, No. 1, 167--209 (2012; Zbl 1250.43002)] provided two recent studies in this direction.NEWLINENEWLINEThe present paper gives a refreshingly new approach, defining Beurling-Fourier algebras without explicitly defining what a weight is! Instead, the crucial notion is that of the inverse of a weight. This is an element \(\omega^{-1}\) of the multiplier algebra of the reduced \(C^*_r(G)\) with certain special properties. The corresponding Beurling-Fourier algebra is defined by \( A(G,\omega) = \{ \omega^{-1}f: f\in A(G)\}\). The authors show that, for abelian groups, these ``are in perfect duality with the classical Beurling algebras'' and that this approach subsumes those of the earlier papers when \(G\) is compact. Another merit of the present approach is that it easily extends to yield a definition of Beurling-Figà-Talamanca-Herz algebras \(A_p(G,\omega), 1<p<\infty\). (The classical Figà-Talamanca-Herz algebras \(A_p(G)\) reduce to the Fourier algebra when \(p=2\) and \(A_2(G,\omega) = A(G,\omega)\).) The theory of \(p\)-operator spaces is used for this purpose. The main result about these algebras is that \(G\) is amenable if and only if some \(A_p(G,\omega)\) has a bounded approximate identity.