Abstract
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.
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This work is supported by NNSFC (10071046), PNSFS (981009), PYSFS (20031009) and China Postdoctoral Science Foundation
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Cui, J.L., Hou, J.C. A Characterization of Homomorphisms Between Banach Algebras. Acta Math Sinica 20, 761–768 (2004). https://doi.org/10.1007/s10114-004-0312-8
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DOI: https://doi.org/10.1007/s10114-004-0312-8