Endomorphisms of an extremal algebra

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.