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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Endomorphisms of an extremal algebra


Authors: Herbert Kamowitz and Dennis Wortman
Journal: Proc. Amer. Math. Soc. 104 (1988), 852-858
MSC: Primary 47B38; Secondary 39B70, 46J99
MathSciNet review: 931733
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Abstract: Let $ Ea[ - 1,1]$ denote the extremal algebra on $ [ - 1,1]$ as defined in Bonsall and Duncan, Numerical ranges. II. We show that every nonzero endomorphism $ T$ on $ Ea[ - 1,1]$ has the form $ Tf(x) \to f(Ax + B)$ where $ A$ and $ B$ are real and $ \vert A\vert + \vert B\vert \leq 1$. Further, the endomorphism $ T$ is an automorphism if, and only if, $ B = 0$ and $ A = 1$ or $ -1$, while $ T$ is a nonzero compact endomorphism if, and only if, $ T:f(x) \to f(B)$ for some $ B$ in $ [ - 1,1]$. Also included in this note are several results related to compact endomorphisms of regular commutative semisimple Banach algebras.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0931733-0
PII: S 0002-9939(1988)0931733-0
Article copyright: © Copyright 1988 American Mathematical Society