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

MathSciNet review:
2957207

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Abstract: Let be a (complex, unital) semisimple Banach algebra and denote by its open spectral unit ball, that is, the set

where denotes the spectrum of in and is the open unit disc in the complex plane. We prove that if is a holomorphic map satisfying and (the identity of ), then for in the intersection of all closed discs lying inside and containing equals the intersection of all closed discs lying inside and containing . When all the elements of have an at most countable spectrum and is biholomorphic, this implies that preserves the convex hull of the spectrum. As an application of the same equality, we prove that if is a semisimple Banach algebra and is a unital surjective spectral isometry, then in the case when 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

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.