Nonlinear maps on von Neumann algebras preserving the star order

摘要

Star order is defined on a C*-algebra in the following way: a {precedes above single-line equals sign} b if a*a = a*b and aa* = ba*. Let A{script} be a von Neumann algebra without Type I_2 direct summand. Let A{script}n be the set of all normal elements of A{script}. Suppose that φ{symbol}: A{script}n → A{script}~n is a continuous bijection that preserves the star order on A{script}~n in both directions. Further, let there is a function f: ? → ? and an invertible central element c in A{script} such that φ{symbol}(λ1) = f(λ)c for all λ ∈ ?. We show that there is a unique Jordan *-isomorphism ψ: A{script} → A{script} such that Ramifications of this result as well as optimality of the assumptions are discussed.