Invariant subspaces in the dual of $A_{cb}(G)$ and $A_M (G)$

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 $AM(G)$ or $A_{cb}(G)$, then $UCB(\mathcal{A}(G)) \subseteq \mathcal{A}P(A(G))$ if and only if $G$ is discrete. We also show that if $UCB(\mathcal{A}(G)) = \mathcal{A}(G)^\ast$, 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 $\mathcal{A}(G)^\ast$ that contains $L^1(G)$, then $i^\ast (X)$ is closed in $A(G)$ if and only if $G$ is amenable.

Keywords

Fourier algebra, multipliers, Arens regularity, uniformly continuous functionals, topologically invariant means

2010 Mathematics Subject Classification

Primary 43A07, 43A22, 46J10. Secondary 47L25.

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The research of Brian Forrest is supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

Received 12 April 2023

Accepted 31 May 2023

Published 26 July 2023