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A class of spectral sets

Author: C. Robert Warner
Journal: Proc. Amer. Math. Soc. 57 (1976), 99-102
MSC: Primary 43A45; Secondary 46J20
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.

References [Enhancements On Off] (What's this?)

  • [1] A. P. Calderón, Ideals in group algebras, Sympos. on Harmonic Analysis and Related Integral Transforms, Cornell University, 1956 (mimeographed).
  • [2] C. S. Herz, The spectral theory of bounded functions, Trans. Amer. Math. Soc. 94 (1960), 181-232. MR 24 #A1627. MR 0131779 (24:A1627)
  • [3] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der math. Wissenschaften, Band 152, Springer-Verlag, New York and Berlin, 1970. MR 41 #7378; erratum, 42, p. 1825. MR 0262773 (41:7378)
  • [4] J.-P. Kahane and R. Salem, Ensembles parfaits et séries trigonométriques, Actualités Sci. Indust., no. 1301, Hermann, Paris, 1963. MR 28 #3279. MR 0160065 (28:3279)
  • [5] H. Reiter, Classical harmonic analysis and locally compact groups, Oxford Univ. Press, Oxford, 1968. MR 46 #5933. MR 0306811 (46:5933)
  • [6] -, Contributions to harmonic analysis. VI, Ann. of Math. (2) 77 (1963), 522-562. MR 27 #1778. MR 0151795 (27:1778)
  • [7] W. Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Appl. Math., no. 12, Interscience, New York, 1962. MR 27 #2808. MR 0152834 (27:2808)
  • [8] S. Saeki, Spectral synthesis for the Kronecker sets, J. Math. Soc. Japan 21 (1969), 549-563. MR 40 #7733. MR 0254525 (40:7733)
  • [9] C. R. Warner, A generalization of the Šilov-Wiener Tauberian theorem, J. Functional Analysis 4 (1969), 329-331. MR 40 #649. MR 0247381 (40:649)

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Additional Information

Keywords: Spectral synthesis, Ditkin sets, $ C$-sets, Ditkin's condition, regular Banach algebras
Article copyright: © Copyright 1976 American Mathematical Society

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