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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.

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Keywords: Extreme point, spectral carrier, spectral decomposition, facial ideal, chain of idempotents, simplex
Article copyright: © Copyright 1981 American Mathematical Society

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