On Fourier algebra homomorphisms
For any locally compact group \(G\), its coset ring is defined to be the ring of subsets of \(G\) generated by the left cosets of open subgroups of \(G\). If \(G\) and \(H\) are locally compact groups and \(Y\) lies in the coset ring of \(H\), a piecewise affine map \(\alpha: Y \to G\) is a map which is piecewise defined as left translates of continuous group homomorphisms from open subgroups of \(H\). Let \(A(G)\) and \(B(H)\) denote the Fourier algebra of \(G\) and the Fourier-Stieltjes algebra of \(H\), respectively. For abelian \(G\) and \(H\) -- so that \(A(G) \cong L^1(\widehat{G})\) and \(B(H) \cong M(\widehat{H})\) --, \textit{P. J. Cohen} showed that the (automatically bounded) algebra homomorphisms from \(A(G)\) to \(B(H)\) arise through composition with piecewise affine maps from sets in the coset ring of \(H\) into \(G\) [Am. J. Math. 82, 191--212 (1960; Zbl 0099.25504)]. Later, \textit{B. Host} extended Cohen's result to homomorphisms from \(A(G)\) to \(B(H)\), where only \(G\) had to be abelian [Bull. Soc. Math. Fr. 114, 215--223 (1986; Zbl 0606.43002)]. He also observed that his result still holds true if \(G\) had an abelian subgroup of finite index, but that it becomes false for all other \(G\): if \(G\) does not have an abelian subgroup of finite index, then the homomorphism \(A(G) \ni f \mapsto \check{f}\) is not of the described kind. In the paper under review, the author obtains an extension of the Cohen-Host result in a novel and very interesting direction. As preduals of von Neumann algebras, \(A(G)\) and \(B(H)\) both have canonical operator space structures, so that it makes sense to speak of completely bounded maps between \(A(G)\) and \(B(H)\). The author proves that, if \(G\) is a discrete, amenable group and if \(H\) is any locally compact group, then every completely bounded homomorphism from \(A(G)\) to \(B(H)\) is obtained through composition with a piecewise affine map. (In a more recent paper, written in collaboration with \textit{N. Spronk}, she has even shown that the discreteness hypothesis for \(G\) can be dropped [Completely bounded homomorphisms of the Fourier algebras. J. Funct. Anal., to appear].) The paper concludes with a discussion of the ranges of algebra homomorphisms between Fourier algebras in the spirit of (and extending) earlier work by \textit{A.\ Kepert} [Can. Math. Bull. 40, No. 2, 183--192 (1997; Zbl 0888.43001)].
scientific article; zbMATH DE number 2105324