Spectrally bounded operators on simple -algebras

Author:
Martin Mathieu

Journal:
Proc. Amer. Math. Soc. **132** (2004), 443-446

MSC (2000):
Primary 47B48; Secondary 46L05, 47A65, 17C65

DOI:
https://doi.org/10.1090/S0002-9939-03-07215-0

Published electronically:
August 7, 2003

MathSciNet review:
2022367

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A linear mapping from a subspace of a Banach algebra into another Banach algebra is called spectrally bounded if there is a constant such that for all , where denotes the spectral radius. We prove that every spectrally bounded unital operator from a unital purely infinite simple -algebra onto a unital semisimple Banach algebra is a Jordan epimorphism.

**1.**B. Aupetit and M. Mathieu,*The continuity of Lie homomorphisms*, Studia Math.**138**(2000), 193-199. MR**2001a:46050****2.**L. G. Brown and G. K. Pedersen,*-algebras of real rank zero*, J. Funct. Anal.**99**(1991), 131-149. MR**92m:46086****3.**J. Cuntz,*The structure of multiplication and addition in simple**-algebras*, Math. Scand.**40**(1977), 215-233. MR**58:17862****4.**K. R. Davidson,*-algebras by example*, Fields Inst. Monographs**6**, Amer. Math. Soc., Providence, RI, 1996. MR**97i:46095****5.**T. Fack,*Finite sums of commutators in**-algebras*, Ann. Inst. Fourier, Grenoble**32**(1982), 129-137. MR**83g:46051****6.**L. W. Marcoux,*On the linear span of the projections in certain simple**-algebras*, Indiana Univ. Math. J.**51**(2002), 753-771.**7.**M. Mathieu,*Spectrally bounded traces on**-algebras*, Bull. Austral. Math. Soc.**68**(2003), 169-173.**8.**M. Mathieu and G. J. Schick,*First results on spectrally bounded operators*, Studia Math.**152**(2002), 187-199. MR**2003e:47066****9.**M. Mathieu and G. J. Schick,*Spectrally bounded operators from von Neumann algebras*, in press, J. Operator Theory (2003).**10.**M. Mathieu and A. R. Sourour,*Hereditary properties of spectral isometries*, Archiv Math. (Basel), to appear.**11.**C. Pop,*Finite sums of commutators*, Proc. Amer. Math. Soc.**130**(2002), 3039-3041. MR**2003c:46070****12.**M. Rørdam,*A simple**-algebra with a finite and an infinite projection*, preprint.**13.**S. Zhang,*A property of purely infinite simple**-algebras*, Proc. Amer. Math. Soc.**109**(1990), 717-720. MR**90k:46134**

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Additional Information

**Martin Mathieu**

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

Email:
m.m@qub.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-03-07215-0

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