Norm One Idempotent $cb$-Multipliers with Applications to the Fourier Algebra in the $cb$-Multiplier Norm

Canadian mathematical bulletin

Brian E. Forrest¹  Volker Runde² 

[1] Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1;Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1

关键词: amenability;    bounded approximate identity;    $cb$-multiplier norm;    Fourier algebra;    norm one idempotent;   

DOI  :  10.4153/CMB-2011-098-0

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely bounded multipliers of $A(G)$, and let $A_{mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E subset G$ is a norm one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of $A_{mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize those $G$ for which $A_{mathit{Mcb}}(G)$ is $1$-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.)

【 授权许可】

Unknown