Inductive extreme non-Arens regularity of the Fourier algebra A(G)
Let G be a non-discrete locally compact group, A(G) the Fourier algebra of G, VN(G) the von Neumann algebra generated by the left regular representation of G which is identified with A(G)∗, and WAP(Ĝ) the space of all weakly almost periodic functionals on A(G). We show that there exists a directed family H of open subgroups of G such that: (1) for each H ∈ H, A(H) is extremely non-Arens regular; (2) VN(G) = ⋃ H∈H VN(H) and VN(G)/WAP(Ĝ) = ⋃ H∈H[VN(H)/WAP(Ĥ)]; (3) A(G) = ⋃ H∈HA(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 ∈ H} of Fourier algebras has a kind of inductively compatible extreme nonArens regularity.