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

 

A Cartan theorem for Banach algebras


Author: Thomas Ransford
Journal: Proc. Amer. Math. Soc. 124 (1996), 243-247
MSC (1991): Primary 46Hxx; Secondary 32Hxx
MathSciNet review: 1307559
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Abstract: Let $A$ be a semisimple Banach algebra, and let $\Omega_A$ be its spectral unit ball. We show that every holomorphic map $G\colon \Omega_A\to \Omega_A$ satisfying $G(0)=0$ and $G'(0)=I$ fixes those elements of $\Omega_A$ which belong to the centre of $A$, but not necessarily any others. Using this, we deduce that the automorphisms of $\Omega_A$ all leave the centre invariant. As a further application, we give a new proof of Nagasawa's generalization of the Banach-Stone theorem.


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Additional Information

Thomas Ransford
Affiliation: Département de Mathématiques et de Statistique, Université Laval, Québec (QC), Canada G1K 7P4
Email: ransford@mat.ulaval.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03197-8
PII: S 0002-9939(96)03197-8
Received by editor(s): August 24, 1994
Additional Notes: The author was supported by grants from NSERC and FCAR
Communicated by: Theodore Gamelin
Article copyright: © Copyright 1996 American Mathematical Society