Strong Ditkin sets in the spectrum of a commutative Banach algebra and the Fourier algebra

We extend the notion of a strong Ditkin set in the dual group for the \({L^1}\)-algebra of a locally compact abelian group as well as a large number of results for such sets to the setting of a general regular and semisimple commutative Banach algebra and its spectrum. In particular, we study various stability and inheritance properties. Moreover, we present some applications to Fourier algebras of locally compact groups and an example of a compact, infinite double coset hypergroup for which every closed subset is a strong Ditkin set for its Fourier algebra.