Idempotents of small norm

Abstract

Let $\Gamma$ be a locally compact group. We answer two questions left open in [7] and [9]: i) For abelian $\Gamma$, we prove that if ${\chi }_S\in B(\Gamma )$ is an idempotent with norm $\Vert {\chi }_S\Vert <\frac{4}{3}$, then $S$ is the union of two cosets of an open subgroup of $\Gamma$. ii) For general $\Gamma$, we prove that if ${\chi }_S\in M_{cb}A(\Gamma )$ is an idempotent with norm ${\Vert {\chi }_S\Vert }_{cb}<\frac{1+\sqrt{2}}{2}$, then $S$ is an open coset in $\Gamma$.

5 pages