Idempotents of large norm and homomorphisms of Fourier algebras

TL;DR

This paper characterizes when large-norm idempotents exist in Fourier and Fourier-Stieltjes algebras of locally compact groups, linking their existence to the presence of high-norm homomorphisms between these algebras.

Contribution

It establishes necessary and sufficient conditions for large-norm idempotents and connects their existence to high-norm homomorphisms between Fourier algebras.

Findings

Existence of large-norm idempotents in B(G) implies large-norm homomorphisms from A(H) to B(G).

Existence of large-norm homomorphisms from A(H) to B(G) implies large-norm idempotents in B(G) for some amenable H.

Conditions for large-norm idempotents are characterized precisely in terms of the group structure.

Abstract

We provide necessary and sufficient conditions for the existence of idempotents of arbitrarily large norms 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).