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.

Keywords:

• star order
• nonlinear maps preserving the star order
• C*-algebra
• von Neumann algebra

AMS Subject Classifications::

• 06A06
• 46L10
• 46L40

Acknowledgements

This work was supported by the Grant Agency of the Czech Republic, Grant no. P201/12/0290 and by the Grant Agency of the Czech Technical University in Prague, Grant no. SGS10/264/OHK3/3T/13.