In this paper, we discuss some multiplicative preservers and give some characterizations of isomorphisms or conjugate isomorphisms on β(X), where β(X) denotes the algebra of all bounded linear operators on a Banach space X.
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.
