Operator Segal algebras in Fourier algebras
Tom 179 / 2007
Streszczenie
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.
