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

Abdul Khan; Mohd Raza; Husain Alhazmi

doi:10.15672/hujms.1311207

Research Article

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

Year 2025, Volume: 54 Issue: 2, 378 - 388, 28.04.2025

Abdul Khan , Mohd Raza , Husain Alhazmi

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

Abstract

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

Keywords

prime ring, semiprime ring, normal ring, involution, left ∗-centralizer, Jordan left ∗-centralizer, generalized derivation

Supporting Institution

Deanship of Scientific Research, King Abdulaziz University, Saudi Arabia

Project Number

G-212-662-1441

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