Uniqueness of complete norms for quotients of Banach function algebras

We prove that every quotient algebra of a unital Banach function algebra A has a unique complete norm if A is a Ditkin algebra. The theorem applies, for example, to the algebra A (Γ) of Fourier transforms of the group algebra ${\displaystyle L^1\left(G\right)}$ of a locally compact abelian group (with identity adjoined if Γ is not compact). In such algebras non-semisimple quotients ${\displaystyle A\frac{\Gamma }{\overline{J\left(E\right)}}}$ arise from closed subsets E of Γ which are sets of non-synthesis. Examples are given to show that the condition of Ditkin cannot be relaxed. We construct a variety of mutually non-equivalent norms for quotients of the Mirkil algebra M, which fails Ditkin's condition at only one point of ${\displaystyle {\Phi }_M}$.