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A condition for spectral continuity of positive elements

Author: S. Mouton
Journal: Proc. Amer. Math. Soc. 137 (2009), 1777-1782
MSC (2000): Primary 46H05, 47A10
Published electronically: November 4, 2008
MathSciNet review: 2470837
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Abstract: Let $a$ be an element of a Banach algebra $A$. We introduce a compact subset $T(a)$ of the complex plane, show that the function which maps $a$ onto $T(a)$ is upper semicontinuous and use this fact to provide a condition on $a$ which ensures that if $(a_n)$ is a sequence of positive elements converging to $a$, then the sequence of the spectral radii of the terms $a_n$ converges to the spectral radius of $a$ in the case that $A$ is partially ordered by a closed and normal algebra cone and $a$ is a positive element.

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Additional Information

S. Mouton
Affiliation: Department of Mathematical Sciences, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa

Keywords: Ordered Banach algebra, positive element, spectrum, upper semicontinuity.
Received by editor(s): June 29, 2007
Received by editor(s) in revised form: April 22, 2008, and July 21, 2008
Published electronically: November 4, 2008
Additional Notes: The author thanks the referee for making useful suggestions.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.