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Spectrally bounded operators on simple $C^{*}$-algebras

Author: Martin Mathieu
Journal: Proc. Amer. Math. Soc. 132 (2004), 443-446
MSC (2000): Primary 47B48; Secondary 46L05, 47A65, 17C65
Published electronically: August 7, 2003
MathSciNet review: 2022367
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Abstract: A linear mapping $T$ from a subspace $E$ of a Banach algebra into another Banach algebra is called spectrally bounded if there is a constant $M\geq 0$such that $r(Tx)\leq M\,r(x)$ for all $x\in E$, where $r(\,\cdot \,)$ denotes the spectral radius. We prove that every spectrally bounded unital operator from a unital purely infinite simple $C^*$-algebra onto a unital semisimple Banach algebra is a Jordan epimorphism.

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Additional Information

Martin Mathieu
Affiliation: Department of Pure Mathematics, Queen’s University Belfast, Belfast BT7 1NN, Northern Ireland

Keywords: Spectrally bounded operators, Jordan homomorphisms, purely infinite simple $C^*$-algebras
Received by editor(s): September 30, 2002
Published electronically: August 7, 2003
Additional Notes: This paper was written during a visit to the Departamento de Análisis Matemático de la Universidad de Granada, Granada, Spain. The author gratefully acknowledges the generous hospitality extended to him by his colleagues there. The paper is part of the research carried out in the EC network Analysis and Operators (HPRN-CT-2000-00116)
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society

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