Nonlinear maps on von Neumann algebras preserving the star order

Abstract

Star order is defined on a C*-algebra in the following way: a ⪯ b if a*a = a*b and aa* = ba*. Let 𝒜 be a von Neumann algebra without Type I2 direct summand. Let 𝒜 n be the set of all normal elements of 𝒜. Suppose that ϕ: 𝒜 n → 𝒜 n is a continuous bijection that preserves the star order on 𝒜 n in both directions. Further, let there is a function f : ℂ → ℂ and an invertible central element c in 𝒜 such that ϕ(λ1) = f(λ)c for all λ ∈ ℂ. We show that there is a unique Jordan *-isomorphism ψ: 𝒜 → 𝒜 such that Ramifications of this result as well as optimality of the assumptions are discussed.