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 denote the extremal algebra on as defined in Bonsall and Duncan, *Numerical ranges*. II. We show that every nonzero endomorphism on has the form where and are real and . Further, the endomorphism is an automorphism if, and only if, and or , while is a nonzero compact endomorphism if, and only if, for some in . Also included in this note are several results related to compact endomorphisms of regular commutative semisimple Banach algebras.

**[1]**F. F. Bonsall and J. Duncan,*Numerical ranges. II*, Cambridge University Press, New York-London, 1973. London Mathematical Society Lecture Notes Series, No. 10. MR**0442682****[2]**Yngve Domar,*On the Banach algebra 𝐴(𝐺) for smooth sets Γ⊂𝑅ⁿ*, Comment. Math. Helv.**52**(1977), no. 3, 357–371. MR**0477603****[3]**Yitzhak Katznelson,*An introduction to harmonic analysis*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0248482****[4]**Allan M. Sinclair,*The Banach algebra generated by a hermitian operator*, Proc. London Math. Soc. (3)**24**(1972), 681–691. MR**0305068**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1988-0931733-0

Article copyright:
© Copyright 1988
American Mathematical Society