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.
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