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

 

On the automorphisms of the spectral unit ball


Author: Constantin Costara
Journal: Proc. Amer. Math. Soc. 140 (2012), 4181-4186
MSC (2010): Primary 46Hxx; Secondary 32Hxx, 47A10
Published electronically: April 2, 2012
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Abstract: Let $ \mathcal {A}$ be a (complex, unital) semisimple Banach algebra and denote by $ \Omega _{\mathcal {A}}$ its open spectral unit ball, that is, the set

$\displaystyle \Omega _{\mathcal {A}}=\{a\in \mathcal {A}:\sigma \left ( a\right ) \subseteq \mathbf {D}\},$    

where $ \sigma \left ( a\right ) $ denotes the spectrum of $ a$ in $ \mathcal {A}$ and $ \mathbf {D}$ is the open unit disc in the complex plane. We prove that if $ F:\Omega _{\mathcal {A}}\rightarrow \Omega _{\mathcal {A}}$ is a holomorphic map satisfying $ F\left ( 0\right ) =0$ and $ F^{\prime }\left ( 0\right ) =I$ (the identity of $ \mathcal {A}$), then for $ a$ in $ \Omega _{ \mathcal {A}}$ the intersection of all closed discs lying inside $ \mathbf {D}$ and containing $ \sigma \left ( a\right ) $ equals the intersection of all closed discs lying inside $ \mathbf {D}$ and containing $ \sigma \left ( F\left ( a\right ) \right ) $. When all the elements of $ \mathcal {A}$ have an at most countable spectrum and $ F$ is biholomorphic, this implies that $ F$ preserves the convex hull of the spectrum. As an application of the same equality, we prove that if $ \mathcal {B}$ is a semisimple Banach algebra and $ T: \mathcal {A } \rightarrow \mathcal {B}$ is a unital surjective spectral isometry, then $ \sigma \left ( T\left ( a\right ) \right ) =\sigma \left ( a\right ) $ in the case when $ \sigma \left ( a\right ) $ has exactly two elements.

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Additional Information

Constantin Costara
Affiliation: Faculty of Mathematics and Informatics, Ovidius University, Mamaia Boulevard 124, 900527 Constanţa, Romania
Email: cdcostara@univ-ovidius.ro

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11266-3
PII: S 0002-9939(2012)11266-3
Keywords: Spectrum, spectral unit ball, holomorphic mappings
Received by editor(s): January 15, 2011
Received by editor(s) in revised form: May 18, 2011
Published electronically: April 2, 2012
Additional Notes: This work was supported by CNCSIS-UEFISCSU, project number 24/06.08.2010, PN II-RU Code 300/2010.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.