Invariant Subspaces in the Dual of $A_{cb}(G)$ and $A_M(G)$

Abstract: Let $G$ be a locally compact group. In this paper, we study various invariant subspaces of the duals of the algebras $A_M(G)$ and $A_{cb}(G)$ obtained by taking the closure of the Fourier algebra $A(G)$ in the multiplier algebra $MA(G)$ and completely bounded multiplier algebra $M_{cb}A(G)$ respectively. In particular, we will focus on various functorial properties and containment relationships between these various invariant subspaces including the space of uniformly continuous functionals and the almost periodic and weakly almost periodic functionals. Amongst other results, we show that if $\mathcal{A}(G)$ is either $A_M(G)$ or $A_{cb}(G)$, then $UCB(\mathcal{A}(G))\subseteq WAP(G)$ if and only if $G$ is discrete. We also show that if $UCB(\mathcal{A}(G))=\mathcal{A}(G)^*$, then every amenable closed subgroup of $G$ is compact. Let $i:A(G)\to \mathcal{A}(G)$ be the natural injection. We show that if $X$ is any closed topologically introverted subspace of $\mathcal{A}(G)^*$ that contains $L^1(G)$, then $i^*(X)$ is closed in $A(G)$ if and only if $G$ is amenable.