ORIGINAL ARTICLES

Acta Mathematica Sinica

, Volume 20, Issue 4, pp 761-768

A Characterization of Homomorphisms Between Banach Algebras

  • Jian Lian CuiAffiliated withSchool of Mathematical Sciences, Peking UniversityDepartment of Applied Mathematics, Taiyuan University of Technology Email author 
  • , Jin Chuan HouAffiliated withDepartment of Mathematics, Shanxi Teachers UniversityDepartment of Mathematics, Shanxi University

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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.

Keywords

Banach algebras Homomorphisms Jordan homomorphisms

MR (2000) Subject Classification

46L10 47B48 47B49