Measurable schur multipliers and completely bounded multipliers of the Fourier algebras

Abstract

Let $G$ be a locally compact group, $\mathrm{L}^p (G)$ be the usual $\mathrm{L}^p$-space for $1 \leq p \leq \infty$, and $\mathrm{A} (G)$ be the Fourier algebra of $G$. Our goal is to study, in a new abstract context, the completely bounded multipliers of $\mathrm{A} (G)$, which we denote $\mathrm{M_{cb}A} (G)$. We show that $\mathrm{M_{cb}A} (G)$ can be characterised as the ‘invariant part’ of the space of (completely) bounded normal $\mathrm{L}^\infty (G)$-bimodule maps on $\mathcal{B}(\mathrm{L}^2 (G))$, the space of bounded operators on $\mathrm{L}^2 (G)$. In doing this we develop a function-theoretic description of the normal $\mathrm{L}^\infty (X, \mu)$-bimodule maps on $\mathcal{B} (\mathrm{L}^2 (X, \mu))$, which we denote by $\mathrm{V}^\infty (X, \mu)$, and name the {\it measurable Schur multipliers} of $(X, \mu)$. Our approach leads to many new results, some of which generalise results hitherto known only for certain classes of groups. Those results which we develop here are a uniform approach to obtaining the functorial properties of $\mathrm{M_{cb} A} (G)$, and a concrete description of a standard predual of $\mathrm{M_{cb} A} (G)$.