A class of spectral sets

Abstract:

The two main results are: (i) If the union and intersection of two closed sets are Ditkin sets, then each of the sets is a Ditkin set. (ii) If the union of two sets is a spectral set and their intersection is a Ditkin set, then each of the sets is a spectral set. A corollary of (i) is a generalization of a theorem due to Calderón which proved that closed polyhedral sets in ${R^n}$ are Ditkin (= Calderón) sets. A corollary of (ii) establishes an analogous result for spectral sets. The proofs hold for commutative semisimple regular Banach algebras which satisfy Ditkin’s condition-that the empty set and singletons are Ditkin sets in the maximal ideal space.