On the automorphisms of the spectral unit ball
Author:
Constantin Costara
Journal:
Proc. Amer. Math. Soc. 140 (2012), 41814186
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@univovidius.ro
DOI:
http://dx.doi.org/10.1090/S000299392012112663
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 CNCSISUEFISCSU, project number 24/06.08.2010, PN IIRU 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.
