A Characterization of Homomorphisms Between Banach Algebras
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- Cui, J.L. & Hou, J.C. Acta Math Sinica (2004) 20: 761. doi:10.1007/s10114-004-0312-8
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We show that every unital invertibility preserving linear map from a von Neumann algebra onto a semi-simple Banach algebra is a Jordan homomorphism; this gives an affirmative answer to a problem of Kaplansky for all von Neumann algebras. For a unital linear map Φ from a semi-simple complex Banach algebra onto another, we also show that the following statements are equivalent: (1) Φ is an homomorphism; (2) Φ is completely invertibility preserving; (3) Φ is 2-invertibility preserving.