Embedding Nilpotent Finite Dimensional Banach Algebras into Amenable Banach Algebras

We prove that any finite dimensional nilpotent commutative Banach algebra can be embedded isometrically into a commutative amenable Banach algebra, whose approximate diagonal is bounded by one. This amenable algebra is constructed by means of a quotient of a Fourier algebra by a closed ideal, whose hull is a set of non-synthesis.