, Volume 20, Issue 4, pp 761-768

A Characterization of Homomorphisms Between Banach Algebras

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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.

This work is supported by NNSFC (10071046), PNSFS (981009), PYSFS (20031009) and China Postdoctoral Science Foundation