Completely bounded bimodule maps and spectral synthesis

TL;DR

This paper explores the relationship between spectral synthesis properties of subsets in a locally compact group and their Cartesian products within the framework of completely bounded multipliers and the Haagerup tensor product, revealing new duality results.

Contribution

It establishes a connection between spectral synthesis of subsets in a group and their product sets in the Haagerup tensor product, and characterizes invariant bimodule maps in terms of support on the antidiagonal.

Findings

Spectral synthesis of E^{lat} implies local spectral synthesis of E.

For Moore groups, spectral synthesis of E implies spectral synthesis of E^{lat}.

Invariant bimodule maps supported on the antidiagonal are characterized in weakly amenable groups.

Abstract

We initiate the study of the completely bounded multipliers of the Haagerup tensor product $A(G)\otimes_{\rm h} A(G)$ of two copies of the Fourier algebra $A(G)$ of a locally compact group $G$. If $E$ is a closed subset of $G$ we let $E^{\sharp} = \{(s,t) : st\in E\}$ and show that if $E^{\sharp}$ is a set of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$ then $E$ is a set of local spectral synthesis for $A(G)$. Conversely, we prove that if $E$ is a set of spectral synthesis for $A(G)$ and $G$ is a Moore group then $E^{\sharp}$ is a set of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$. Using the natural identification of the space of all completely bounded weak* continuous $VN(G)'$-bimodule maps with the dual of $A(G)\otimes_{\rm h} A(G)$, we show that, in the case $G$ is weakly amenable, such a map leaves the multiplication algebra of $L^{\infty}(G)$ invariant if and only if its support is contained in the antidiagonal of $G$.