A Cartan theorem for Banach algebras
HTML articles powered by AMS MathViewer
- by Thomas Ransford PDF
- Proc. Amer. Math. Soc. 124 (1996), 243-247 Request permission
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
- Bernard Aupetit, A primer on spectral theory, Universitext, Springer-Verlag, New York, 1991. MR 1083349, DOI 10.1007/978-1-4612-3048-9
- B. Aupetit and H. du T. Mouton, Spectrum preserving linear mappings in Banach algebras, Studia Math. 109 (1994), no. 1, 91–100. MR 1267714, DOI 10.4064/sm-109-1-91-100
- Hari Bercovici, Ciprian Foias, and Allen Tannenbaum, A spectral commutant lifting theorem, Trans. Amer. Math. Soc. 325 (1991), no. 2, 741–763. MR 1000144, DOI 10.1090/S0002-9947-1991-1000144-9
- Masao Nagasawa, Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, K\B{o}dai Math. Sem. Rep. 11 (1959), 182–188. MR 121645
- T. J. Ransford and M. C. White, Holomorphic self-maps of the spectral unit ball, Bull. London Math. Soc. 23 (1991), no. 3, 256–262. MR 1123334, DOI 10.1112/blms/23.3.256
Additional Information
- Thomas Ransford
- Affiliation: Département de Mathématiques et de Statistique, Université Laval, Québec (QC), Canada G1K 7P4
- MR Author ID: 204108
- Email: ransford@mat.ulaval.ca
- Received by editor(s): August 24, 1994
- Additional Notes: The author was supported by grants from NSERC and FCAR
- Communicated by: Theodore Gamelin
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 243-247
- MSC (1991): Primary 46Hxx; Secondary 32Hxx
- DOI: https://doi.org/10.1090/S0002-9939-96-03197-8
- MathSciNet review: 1307559