Linear maps preserving ideals of C*-algebras

Abstract

We show that every unital linear bijection which preserves the maximal left ideals from a semi-simple Banach algebra onto a C ∗ ^{*} -algebra of real rank zero is a Jordan isomorphism. Furthermore, every unital self-adjoint linear bijection on a countably decomposable factor von Neumann algebra is maximal left ideal preserving if and only if it is a *-automorphism.