Idempotents with small norms

Let Γ be a locally compact group. We answer two questions left open in [8] and [10]:(i)For abelian Γ, we prove that if χS∈B(Γ)χS∈B(Γ) is an idempotent with norm ‖χS‖<43, then S is the union of two cosets of an open subgroup of Γ.(ii)For general Γ, we prove that if χS∈McbA(Γ)χS∈McbA(Γ) is an idempotent with norm ‖χS‖cb<1+22, then S is an open coset in Γ.