Idempotents of small norm

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