Idempotents of large norm and homomorphisms of Fourier algebras
Volume 267 / 2022
Studia Mathematica 267 (2022), 109-120
MSC: Primary 43A22; Secondary 43A30.
DOI: 10.4064/sm220111-20-1
Published online: 30 May 2022
Abstract
We provide necessary and sufficient conditions for the existence of idempotents of arbitrarily large norm in the Fourier algebra $A(G)$ and the Fourier–Stieltjes algebra $B(G)$ of a locally compact group $G$. We prove that the existence of idempotents of arbitrarily large norm in $B(G)$ implies the existence of homomorphisms of arbitrarily large norm from $A(H)$ into $B(G)$ for every locally compact group $H$. A partial converse is also obtained: the existence of homomorphisms of arbitrarily large norm from $A(H)$ into $B(G)$ for some amenable locally compact group $H$ implies the existence of idempotents of arbitrarily large norm in $B(G)$.
