Amenability properties of Fourier algebras and Fourier-Stieltjes algebras: a survey

Volume 91 / 2010

Nico Spronk Banach Center Publications 91 (2010), 365-383 MSC: Primary 43-02, 43A30, 46H25, 46L07; Secondary 43A07, 43A20, 43A10, 43A77, 22D10, 46J20, 43A85. DOI: 10.4064/bc91-0-22

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.

Authors

  • Nico SpronkDepartment of Pure Mathematics
    University of Waterloo
    200 University Ave. W.
    Waterloo, Ontario, N2L 3G1, Canada
    e-mail

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