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Characterizing homomorphisms and derivations on C*-algebras

Published online by Cambridge University Press:  24 July 2008

J. Alaminos ,
J. Extremera ,
A. R. Villena  and
M. Brešar
J. Alaminos
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (alaminos@ugr.es; jlizana@ugr.es; avillena@ugr.es)
J. Extremera
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (alaminos@ugr.es; jlizana@ugr.es; avillena@ugr.es)
A. R. Villena
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (alaminos@ugr.es; jlizana@ugr.es; avillena@ugr.es)
M. Brešar
Affiliation:
Department of Mathematics, PEF, University of Maribor, Koroška 160, Slovenia (bresar@uni-mb.si)
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Abstract

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The main theorem states that a bounded linear operator $h$ from a unital $C^{\ast}$-algebra $A$ into a unital Banach algebra $B$ must be a homomorphism provided that $h(\bm{1})=\bm{1}$ and the following condition holds: if $x,y,z\in A$ are such that $xy=yz=0$, then $h(x)h(y)h(z)=0$. This theorem covers various known results; in particular it yields Johnson's theorem on local derivations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 137 , Issue 1 , February 2007 , pp. 1 - 7
Copyright
2007 Royal Society of Edinburgh