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Abstract

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.