Weak spectral synthesis in commutative Banach algebras. III

Let A   be a regular and semisimple commutative Banach algebra with structure space Δ(A)Δ(A). Continuing the investigations of [7] and [9], we establish various results on weak spectral sets in Δ(A)Δ(A). To each closed subset of Δ(A)Δ(A) we associate a descending sequence of subsets of Δ(A)Δ(A) which proves to be a powerful tool in the study of weak spectral synthesis. Applications concern injection type properties, unions of weak spectral sets and projective tensor products. A number of interesting examples is discussed: algebras of m  -times continuously differentiable functions and of Lipschitz functions, and L1(RN)L1(RN).