Idempotents of small norm

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

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