Idempotents of small norm

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 $\left\|\chi_S \right\| < \frac{4}{3}$, 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 $\left\| \chi_S \right\|_{cb} < \frac{1 + \sqrt{2}}{2}$, then $S$ is an open coset in $\Gamma$.