Approximate identity and approximation properties of multidimensional Fourier algebras

TL;DR

This work characterizes approximation and amenability phenomena for multidimensional Fourier algebras $A^n(G)=\otimes_n^{eh}A(G)$. It proves that $G$ is amenable if and only if $A^n(G)$ has a bounded approximate identity and if and only if $A^n(G)$ is operator amenable, extending the classical $A(G)$ results to the multidimensional setting via tensor-product techniques and duality with $VN^n(G)$. The paper then provides a complete AP characterization for $A^n(G)$ within $MA^n(G)$, showing equivalences that involve nets approaching the unit and dual-pairing conditions with sequences in $A^n(G)$, ultimately showing AP transfers to closed subgroups. Finally, it establishes a weak amenability framework for $A^n(G)$ by relating cb-multiplier structures to classical weak amenability of $G$, giving concrete nets in $A(G)^{\otimes n}$ and $M^{cb}(A^n(G))$ with uniform cbm bounds.

Abstract

For a locally compact group $G$, let $A^n(G)$ denote the multidimensional Fourier algebra given by $ \otimes_{n}^{eh} A(G).$ This work explores the approximation identity and operator amenability of the algebra $A^n(G)$. Further, we study the approximation properties (AP) and the concept of weak amenability of the multidimensional Fourier algebra.

Theorems & Definitions (24)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.4
• proof
• Corollary 3.5
• proof