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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
DOI: https://doi.org/10.1090/S0002-9939-1988-0931733-0
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 $|A| + |B| \leq 1$. 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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Article copyright: © Copyright 1988 American Mathematical Society