Non-linear mixed Jordan triple $1$-$*$-product on von Neumann algebras

Year 2024, Early Access, 1 - 11

Abdul Khan , Mohd Raza , Husain Alhazmi

https://doi.org/10.15672/hujms.1311207

Abstract

It is shown that if $\Ma$ and $\Na$ are two von Neumann algebras, one of which has no central abelian projection with $\psi: \Ma \rightarrow \Na$ satisfying mixed Jordan triple $1$-$*$-product, i.e.,
$$\psi(\Aa \circ \Ba \bullet \Ca)=\psi(\Aa) \circ \psi(\Ba) \bullet \psi(\Ca)$$
for all $\Aa, \Ba, \Ca\in \Ma$, then there exists a bijective map $\Psi: \Ma \rightarrow \Na$ such that $\Psi(\Aa)=\psi(\Ia)\psi(\Aa)$ with $\psi(\Ia)^2=\Ia$, whenever $\psi(\Ia)$ is central, and there exsit a central projection $\mathfrak{P} \in \Ma $ such that the restriction of $\psi$ to $\Ma\mathfrak{P}$ is a linear $*$-isomorphism, and to $\Ma(\Ia -\mathfrak{P})$ is a conjugate linear $*$-isomorphism.

Keywords

Mixed Jordan triple $1$-$*$-product, Isomorphism, von Neumann algebra

Supporting Institution

Deanship of Scientific Research, King Abdulaziz University, Saudi Arabia

Project Number

G-212-662-1441

References

• [1] Z. Bai and S. Du, Maps preserving products XY − Y X∗ on von Neumann algebras, J. Math. Anal. Appl. 386, 103-109, 2012