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Abstract

Let $G$ be a locally compact group, and let ${\rm A}(G)$ and ${\rm B}(G)$ denote its Fourier and Fourier-Stieltjes algebras. These algebras are dual objects of the group and measure algebras, ${\rm L}^{-1}(G)$ and ${\rm M}(G)$, in a sense which generalizes the Pontryagin duality theorem on abelian groups. We wish to consider the amenability properties of ${\rm A}(G)$ and ${\rm B}(G)$ and compare them to such properties for ${\rm L}^{-1}(G)$ and ${\rm M}(G)$. For us, “amenability properties” refers to amenability, weak amenability, and biflatness, as well as some properties which are more suited to special settings, such as the hyper-Tauberian property for semisimple commutative Banach algebras. We wish to emphasize that the theory of operator spaces and completely bounded maps plays an indispensable role when studying ${\rm A}(G)$ and ${\rm B}(G)$. We also show some applications of amenability theory to problems of complemented ideals and homomorphisms.