Nonlinear maps preserving Jordan triple η-⁎-products

Abstract

Let η = −1 be a non-zero complex number, and let φ be a not necessarily linear bijection between two von Neumann algebras, one of which has no central abelian projections, satisfying φ(I) = I and preserving the Jordan triple η- * -product. It is showed that φ is a linear * -isomorphism if η is not real and φ is the sum of a linear * -isomorphism and a conjugate linear * -isomorphism if η is real.

Nonlinear mixed Jordan triple *-derivations on factors

“…They were extensively studied because they naturally arise in the problem of representing quadratic functionals with sesquilinear functionals (see [17][18][19]) and in the problem of characterizing ideals (see [2,16]). Particular attention has been paid to understanding maps which preserve the skew Lie product or the Jordan * -product on * -algebras (see [1,3,4,7,9,12,13]).…”

Nonlinear Maps Preserving the Jordan Triple 1- $$*$$ ∗ -Product on Von Neumann Algebras

“…Recently, Huo et al in [5] studied a more general problem. They considered the Jordan triple η- * -product of three elements A, B, and C in a * -algebra A defined by A♦ η B♦ η C = (A♦ η B)♦ η C (we should be aware that ♦ η is not necessarily associative).…”

“…So, the class of those maps preserving the Jordan triple η- * -product is, in principle, wider than the class of maps preserving the Jordan η- * -product. In [5], let η = −1 be a nonzero complex number, and let Φ be a bijection between two von Neumann algebras, one of which has no central abelian projections, satisfying Φ(I) = I and preserving the Jordan triple η- * -product. In [5] it was shown that Φ is a linear * -isomorphism if η is not real and that Φ is the sum of a linear * -isomorphism and a conjugate linear * -isomorphism if η is real.…”

“…We will prove that, if Φ is a bijective map preserving the Jordan triple (−1)- * -product between two von Neumann algebras, one of which has no central abelian projections, then the map Φ(I)Φ is a sum of a linear * -isomorphism and a conjugate linear * -isomorphism, where Φ(I) is a self-adjoint central element in the range with Φ(I) 2 = I. We mention that the methods in [5] do not fit for solving our problem since their proofs heavily depend on the assumption Φ(I) = I.…”

Nonlinear maps preserving the Jordan triple $*$ -product on von Neumann algebras

“…On the one hand, [5] did not include the case η = −1. Obviously, the Jordan (triple) (−1)- * -product is very important and meaningful.…”