Convolutions on compact groups and Fourier algebras of coset spaces

Brian E. Forrest¹ ; Ebrahim Samei² ; Nico Spronk¹

¹Department of Pure Mathematics University of Waterloo Waterloo, ON N2L 3G1, Canada
²Department of Mathematics & Statistics University of Saskatchewan Saskatoon, SK S7N 5E6, Canada

Studia Mathematica, Tome 196 (2010) no. 3, pp. 223-249

Résumé

We study two related questions. (1) For a compact group $G$, what are the ranges of the convolution maps on $\mathrm{A}(G\times G)$ given for $u,v$ in $\mathrm{A}(G)$ by $u\times v\mapsto u\ast\check{v}$ ($\check{v}(s)=v(s^{-1})$) and $u\times v\mapsto u\ast v$? (2) For a locally compact group $G$ and a compact subgroup $K$, what are the amenability properties of the Fourier algebra of the coset space $\mathrm{A}({G/K})$? The algebra $\mathrm{A}({G/K})$ was defined and studied by the first named author.In answering the first question, we obtain, for compact groups which do not admit an abelian subgroup of finite index, some new subalgebras of $\mathrm{A}(G)$. Using those algebras we can find many instances in which $\mathrm{A}({G/K})$ fails the most rudimentary amenability property: operator weak amenability. However, using different techniques, we show that if the connected component of the identity of $G$ is abelian, then $\mathrm{A}({G/K})$ always satisfies the stronger property that it is hyper-Tauberian, which is a concept developed by the second named author. We also establish a criterion which characterises operator amenability of $\mathrm{A}({G/K})$ for a class of groups which includes the maximally almost periodic groups. Underlying our calculations are some refined techniques for studying spectral synthesis properties of sets for Fourier algebras. We even find new sets of synthesis and nonsynthesis for Fourier algebras of some classes of groups.

Détail

DOI : 10.4064/sm196-3-2

Keywords: study related questions compact group what ranges convolution maps mathrm times given mathrm times mapsto ast check check times mapsto ast locally compact group compact subgroup what amenability properties fourier algebra coset space mathrm algebra mathrm defined studied first named author answering first question obtain compact groups which admit abelian subgroup finite index subalgebras mathrm using those algebras many instances which mathrm fails rudimentary amenability property operator weak amenability however using different techniques connected component identity abelian mathrm always satisfies stronger property hyper tauberian which concept developed second named author establish criterion which characterises operator amenability mathrm class groups which includes maximally almost periodic groups underlying calculations refined techniques studying spectral synthesis properties sets fourier algebras even sets synthesis nonsynthesis fourier algebras classes groups