Map preserving mixed skew Lie–Jordan product on von Neumann algebra

Abstract In this paper, we characterized the maps between factor von Neumann algebra that preserves mixed skew Lie–Jordan products [ A , B ] * ◦ C , for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo>,</m:mo> <m:mi>C</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="fraktur">S</m:mi> </m:math> $A,B,C\in \mathfrak{S}$ , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="fraktur">S</m:mi> </m:math> $\mathfrak{S}$ is factor von Neumann algebra and the products are defined like [ A , B ] * = AB − B * A , ( A ◦ B ) = AB + BA . Specifically, we have shown that maps preserving such algebraic structures are linear ∗-isomorphism.