$\mathbb{R}$-linear unitary preservers of C${}^{\ast }$-algebras

Abstract

We study the structure of general $\mathbb{R}$-linear unitary preservers between unital C${}^{\ast }$-algebras. Representing any unital $\mathbb{R}$-linear unitary preserver in terms of three $\mathbb{R}$-linear maps on the self-adjoint part of the domain yields a canonical decomposition into a Jordan-type summand and a purely non-Jordan summand. When the domain is abelian, we completely classify the purely non-Jordan summand via type ${\mathrm{I}}_2$ JW-algebras. For unital C${}^{\ast }$-algebras with no two-dimensional irreducible representations, we characterize a natural class of purely non-Jordan unitary preservers governed by pairs of Jordan $\ast$-homomorphisms.