Type: Article
Publication Date: 1976-01-01
Citations: 10
DOI: https://doi.org/10.1090/s0002-9939-1976-0410275-7
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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.