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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A class of spectral sets


Author: C. Robert Warner
Journal: Proc. Amer. Math. Soc. 57 (1976), 99-102
MSC: Primary 43A45; Secondary 46J20
DOI: https://doi.org/10.1090/S0002-9939-1976-0410275-7
MathSciNet review: 0410275
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0410275-7
Keywords: Spectral synthesis, Ditkin sets, $ C$-sets, Ditkin's condition, regular Banach algebras
Article copyright: © Copyright 1976 American Mathematical Society