Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

A Cartan theorem for Banach algebras

Author(s): 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.

References:

1: B. Aupetit, A primer on spectral theory, Springer-Verlag, New York, 1991. MR 92c:46001
2: B. Aupetit and H. du T. Mouton, Spectrum-preserving linear mappings in Banach algebras, Studia Math. 109 (1994), 91--100. MR 95c:46070
3: H. Bercovici, C. Foias, and A. Tannenbaum, A spectral commutant lifting theorem, Trans. Amer. Math. Soc. 325 (1991), 741--763. MR 91j:47006
4: M. Nagasawa, Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kodai Math. Sem. Rep. 11 (1959), 182--188. MR 22:12379
5: T. J. Ransford and M. C. White, Holomorphic self-maps of the spectral unit ball, Bull. London Math. Soc. 23 (1991), 256--262. MR 92g:32049

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