On Idempotent Measures of Small Norm

In this Master’s Thesis, we set up the groundwork for [8], a paper co-written by the author and Hung Pham. We summarise the Fourier and Fourier-Stieltjes algebras on both abelian and general locally compact groups. Let Г be a locally compact group. We answer two questions left open in [11] and [13]:  1. When Г is abelian, we prove that if ϰs ∈ B(Г) is an idempotent with norm 1 < ||ϰs|| < 4/3 then S is the union of two cosets of an open subgroup of Г.  2. For general Г, we prove that if ϰs ∈ McbA(Г) is an idempotent with norm ||ϰs||cb < 1+√2/2 , then S is an open coset in Г.