Operator Segal algebras in Fourier algebras

Brian E. Forrest¹ ; Nico Spronk¹ ; Peter J. Wood¹

¹ Department of Pure Mathematics University of Waterloo Waterloo, ON, N2L 3G1, Canada

Studia Mathematica, Tome 179 (2007) no. 3, pp. 277-295

Résumé

Let $G$ be a locally compact group, $\mathrm{A}(G)$ its Fourier algebra and $\mathrm{L}^1(G)$ the space of Haar integrable functions on $G$. We study the Segal algebra ${\mathrm{S}^1\!\mathrm{A}(G)}= {\mathrm{A}(G)}\cap{\rm L}^1(G)$ in ${\mathrm{A}(G)}$. It admits an operator space structure which makes it a completely contractive Banach algebra. We compute the dual space of $\mathrm{S}^1\!\mathrm{A}(G)$. We use it to show that the restriction operator $u\mapsto u|_H:{\mathrm{S}^1\!\mathrm{A}(G)}\to{\mathrm{A}(H)}$, for some non-open closed subgroups $H$, is a surjective complete quotient map. We also show that if $N$ is a non-compact closed subgroup, then the averaging operator $\tau_N:{\mathrm{S}^1\!\mathrm{A}(G)}\to \mathrm{L}^1({G/N})$, $\tau_Nu(sN)=\int_N u(sn)\,dn,\!$ is a surjective complete quotient map. This puts an operator space perspective on the philosophy that ${\mathrm{S}^1\!\mathrm{A}(G)}$ is “locally ${\rm A}(G)$ while globally $\mathrm{L}^1$”. Also, using the operator space structure we can show that ${\mathrm{S}^1\!\mathrm{A}(G)}$ is operator amenable exactly when when $G$ is compact; and we can show that it is always operator weakly amenable. To obtain the latter fact, we use E. Samei's theory of hyper-Tauberian Banach algebras.

Détail

DOI : 10.4064/sm179-3-5

Mots-clés : locally compact group mathrm its fourier algebra mathrm space haar integrable functions study segal algebra mathrm mathrm mathrm cap mathrm admits operator space structure which makes completely contractive banach algebra compute dual space mathrm mathrm restriction operator mapsto mathrm mathrm mathrm non open closed subgroups nbsp surjective complete quotient map non compact closed subgroup averaging operator tau mathrm mathrm mathrm tau int surjective complete quotient map puts operator space perspective philosophy mathrm mathrm locally while globally mathrm using operator space structure mathrm mathrm operator amenable exactly when compact always operator weakly amenable obtain latter nbsp sameis theory hyper tauberian banach algebras