Unitary closure and Fourier algebra of a topological group (Q2787150)

This interesting paper is a sequel to an earlier paper by the same authors [Adv. Math. 229, No. 3, 2000--2023 (2012; Zbl 1236.22001)]. In the previous paper, the authors have studied the Fourier-Stieltjes algebra \(B(G)\) of a (not necessarily locally compact) topological group \(G\). Now the authors define and study the Fourier algebra \(A(G)\) of \(G\). A constant difficulty in this as well as in the earlier paper is the lack of a Haar measure on every topological group that is not locally compact.NEWLINENEWLINEThe authors define \(A(G)\) as the intersection of the maximal ideals of \(B(G)\) whose corresponding characters are not in the unitary cover \(\tilde{G}\) of \(G\), where the unitary cover \(\tilde{G}\) of \(G\) is defined as the set of those characters of \(B(G)\) that are unitary in the full group von Neumann algebra \(W^*(G)\), which is the Banach space dual of \(B(G)\). Recall that the notion of Fourier algebras on locally compact groups has been defined by \textit{P. Eymard} in [Bull. Soc. Math. Fr. 92, 181--236 (1964; Zbl 0169.46403)].NEWLINENEWLINEA concept related to the unitary cover is that of the unitary closure \(\bar{G}\) of \(G\), which is defined by the authors as the closure of \(G\) in \(\tilde{G}\) with respect to the relative weak\(^*\) topology of \(W^*(G)\). Using this notion, it is proved in the paper that if \(A(G)\neq \{0\}\), then \(\bar{G}\) must be locally compact and \(A(G)\) must be isometrically isomorphic to \(A(\bar{G})\). Moreover, the authors prove a necessary and sufficient, simple condition for when such a situation occurs. For example, it follows that if \(G\) is a nontrivial (not necessarily closed) subgroup of a locally compact group, then \(A(G)\neq\{0\}\).NEWLINENEWLINEThe paper is then continued with the study of when \(A(G)\) is operator amenable (precisely when the same group \(G\) but equipped with the topology generated by \(B(G)\) is amenable) and when \(A(G)\) is operator weakly amenable (always). For the subclass of [SIN]-groups, the authors also characterise when \(A(G)\) has the weak fixed point property, the Radon-Nikodym property (RNP), the Krein-Milman property (KMP), and the uniform Kadec-Klee property (UKK).NEWLINENEWLINEThe paper is concluded with a list of intriguing open questions.