Nonlinear maps preserves skew products on factor von Neumann Algebra

Let [Formula: see text] and [Formula: see text] be von Neumann algebras. For [Formula: see text], define [Formula: see text], the product ⋆ of [Formula: see text] and [Formula: see text]. This manuscript discusses the bijective mapping [Formula: see text] that preserve the product ⋆ on von Neumann algebras [Formula: see text], for every [Formula: see text]. Especially, we elaborate on the characteristics of the map [Formula: see text]. Precisely, we establish the necessary and sufficient condition that [Formula: see text] is a linear ⋆-ring isomorphism or the negative of a conjugate linear ⋆-isomorphism