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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On spectral theory and convexity


Authors: C. K. Fong and Louisa Lam
Journal: Trans. Amer. Math. Soc. 264 (1981), 59-75
MSC: Primary 46H99; Secondary 46A55, 47B15
MathSciNet review: 597867
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Abstract: A compact convex set $ K$ in a locally convex algebra is said to be a spectral carrier if, for all $ x$, $ y \in K$, we have $ xy = yx \in K$ and $ x + y - xy \in K$. We show that if a compact convex set $ K$ is a spectral carrier, then the idempotents in $ K$ are exactly the extreme points of $ K$ and form a complete lattice. Conversely, if a compact set $ K$ is a closed convex hull of a lattice of commuting idempotents, then $ K$ is a spectral carrier. Furthermore, a metrizable spectral carrier is a Choquet simplex if and only if its extreme points form a chain of idempotents.


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

  • [1] Erik M. Alfsen, Compact convex sets and boundary integrals, Springer-Verlag, New York-Heidelberg, 1971. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57. MR 0445271 (56 #3615)
  • [2] G. Choquet, Lectures on analysis. Vol. II, Benjamin, New York, 1969.
  • [3] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162 (90g:47001a)
  • [4] -, Linear operators. Part III: Spectral operators, Interscience, New York, 1962.
  • [5] I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. MR 0264447 (41 #9041)
  • [6] Leopoldo Nachbin, Topology and order, Translated from the Portuguese by Lulu Bechtolsheim. Van Nostrand Mathematical Studies, No. 4, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1965. MR 0219042 (36 #2125)
  • [7] J. R. Ringrose, On well-bounded operators, J. Austral. Math. Soc. 1 (1959/1960), 334–343. MR 0126167 (23 #A3463)
  • [8] Helmut H. Schaefer, Topological vector spaces, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1966. MR 0193469 (33 #1689)
  • [9] D. R. Smart, Conditionally convergent spectral expansions, J. Austral. Math. Soc. 1 (1959/1960), 319–333. MR 0126166 (23 #A3462)
  • [10] G. T. Whyburn, Topological analysis, Princeton Univ. Press, Princeton, N. J., 1964.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0597867-6
PII: S 0002-9947(1981)0597867-6
Keywords: Extreme point, spectral carrier, spectral decomposition, facial ideal, chain of idempotents, simplex
Article copyright: © Copyright 1981 American Mathematical Society