Structure of Nonlinear Maps Preserving A*B − ξBA onto an Arbitrary ∗-Algebra

We show that every bijection Ψ between complex ∗-algebras satisfying Ψ(A*B−ξBA)=Ψ(A)*Ψ(B)−ξΨ(B)Ψ(A), for a fixed ξ≠0, is additive as soon as its domain is a prime unital ∗-algebra with a nontrivial projection, with nothing assumed about its codomain at all. In the additivity theorems for products built from the involution that we have been able to trace, the codomain carries a hypothesis of the same kind as the domain. But the equation does not merely tolerate an unrestricted codomain: for |ξ|≠1, surjectivity of Ψ alone is already enough to give it a unit and a centre, and once Ψ is bijective it becomes semilinear over that centre, a phenomenon we call range rigidity. This already rules out a surjection onto C0(X) for noncompact X, and, for an infinite dimensional Hilbert space H, onto K(H) or a Schatten class. A second, independent argument then extends the exclusion of C0(X) to every ξ≠0, and reaches the operator ideals on an infinite dimensional H as well at the classical parameter ξ=−1. Through a conjugate-linear equivalence with the classical products AB−ηBA*, the same machinery yields a new proof of the known classification of such maps between factor von Neumann algebras, which is due to Dai and Lu, for every ξ other than −1. At ξ=−1, the classification is the one of Li, Lu and Fang, and it is quoted here rather than reproved.