On spectral theory and convexity
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- by C. K. Fong and Louisa Lam PDF
- Trans. Amer. Math. Soc. 264 (1981), 59-75 Request permission
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
- Erik M. Alfsen, Compact convex sets and boundary integrals, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57, Springer-Verlag, New York-Heidelberg, 1971. MR 0445271 G. Choquet, Lectures on analysis. Vol. II, Benjamin, New York, 1969.
- 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 —, Linear operators. Part III: Spectral operators, Interscience, New York, 1962.
- I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. Translated from the Russian by A. Feinstein. MR 0264447
- Leopoldo Nachbin, Topology and order, Van Nostrand Mathematical Studies, No. 4, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1965. Translated from the Portuguese by Lulu Bechtolsheim. MR 0219042
- J. R. Ringrose, On well-bounded operators, J. Austral. Math. Soc. 1 (1959/1960), 334–343. MR 0126167
- Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
- D. R. Smart, Conditionally convergent spectral expansions, J. Austral. Math. Soc. 1 (1959/1960), 319–333. MR 0126166 G. T. Whyburn, Topological analysis, Princeton Univ. Press, Princeton, N. J., 1964.
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 264 (1981), 59-75
- MSC: Primary 46H99; Secondary 46A55, 47B15
- DOI: https://doi.org/10.1090/S0002-9947-1981-0597867-6
- MathSciNet review: 597867