Inductive extreme non-Arens regularity of the Fourier algebra $A(G)$

Zhiguo Hu¹

¹ Department of Mathematics and Statistics University of Windsor Windsor, Ontario, Canada N9B 3P4

Studia Mathematica, Tome 151 (2002) no. 3, pp. 247-264

Résumé

Let $G$ be a non-discrete locally compact group, $A(G)$ the Fourier algebra of $G$, ${\rm VN}(G)$ the von Neumann algebra generated by the left regular representation of $G$ which is identified with $A(G)^*$, and ${\rm WAP}(\widehat {G})$ the space of all weakly almost periodic functionals on $A(G)$. We show that there exists a directed family ${\cal H}$ of open subgroups of $G$ such that: (1) for each $H \in {\cal H}$, $A(H)$ is extremely non-Arens regular; (2) ${\rm VN}(G) = \bigcup _{H \in {\cal H}} {\rm VN}(H)$ and ${\rm VN}(G)/{\rm WAP}(\widehat {G}) = \bigcup _{H \in {\cal H}} [{\rm VN}(H)/{\rm WAP}(\widehat H)]$; (3) $A(G) = \bigcup _{H \in {\cal H}} A(H)$ and it is a WAP-strong inductive union in the sense that the unions in (2) are strongly compatible with it. Furthermore, we prove that the family $\{ A(H):H \in {\cal H}\} $ of Fourier algebras has a kind of inductively compatible extreme non-Arens regularity.

Détail

DOI : 10.4064/sm151-3-4

Keywords: non discrete locally compact group fourier algebra von neumann algebra generated regular representation which identified * wap widehat space weakly almost periodic functionals there exists directed family cal subgroups each cal extremely non arens regular bigcup cal wap widehat bigcup cal wap widehat bigcup cal wap strong inductive union sense unions strongly compatible furthermore prove family cal fourier algebras has kind inductively compatible extreme non arens regularity