The Mazur–Ulam property for commutative von Neumann algebras

Abstract

ABSTRACTLet be a σ-finite measure space. Given a Banach space X, let the symbol stand for the unit sphere of X. We prove that the space of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm satisfies the Mazur–Ulam property, that is, if X is any complex Banach space, every surjective isometry admits an extension to a surjective real linear isometry . This conclusion is derived from a more general statement which assures that every surjective isometry where K is a Stonean space, admits an extension to a surjective real linear isometry from onto X.