Geodesic
Mappings preserving sum of products $a\circ b+ba^{*}$ on factor von Neumann algebras
Algebra and discrete mathematics, Tome 31 (2021) no. 1, pp. 61-70

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Let $\mathcal{A}$ and $\mathcal{B}$ be two factor von Neumann algebras. In this paper, we proved that a bijective mapping $\Phi \colon\mathcal{A}\to\mathcal{B}$ satisfies $\Phi (a\circ b+ba^{*})=\Phi (a)\circ \Phi (b)+\Phi (b)\Phi (a)^{*}$ (where $\circ $ is the special Jordan product on $\mathcal{A}$ and $\mathcal{B},$ respectively), for all elements $a,b\in \mathcal{A}$, if and only if $\Phi $ is a $\ast $-ring isomorphism. In particular, if the von Neumann algebras $\mathcal{A}$ and $\mathcal{B}$ are type I factors, then $\Phi $ is a unitary isomorphism or a conjugate unitary isomorphism.
Keywords: $\ast$-ring isomorphisms, factor von Neumann algebras. 
@article{ADM_2021_31_1_a3,
     author = {J. C. Ferreira and M. G. B. Marietto},
     title = {Mappings preserving sum of products $a\circ b+ba^{*}$ on factor von {Neumann} algebras},
     journal = {Algebra and discrete mathematics},
     pages = {61--70},
     publisher = {mathdoc},
     volume = {31},
     number = {1},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2021_31_1_a3/}
}
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J. C. Ferreira; M. G. B. Marietto. Mappings preserving sum of products $a\circ b+ba^{*}$ on factor von Neumann algebras. Algebra and discrete mathematics, Tome 31 (2021) no. 1, pp. 61-70. http://geodesic.mathdoc.fr/item/ADM_2021_31_1_a3/