Weak spectral synthesis in commutative Banach algebras

Let A be a semisimple and regular commutative Banach algebra with structure space Δ(A). Generalizing the notion of spectral sets in Δ(A), the considerably larger class of weak spectral sets was introduced and studied in [C.R. Warner, Weak spectral synthesis, Proc. Amer. Math. Soc. 99 (1987) 244–248]. We prove injection theorems for weak spectral sets and weak Ditkin sets and a Ditkin–Shilov type theorem, which applies to projective tensor products. In addition, we show that weak spectral synthesis holds for the Fourier algebra A(G) of a locally compact group G if and only if G is discrete.