$*$-Lie-Jordan-type maps on $C^{*}$-algebras

$*$-Lie-type maps on $C^*$-algebras

“…Let us present some consequences of our main result. The first one provides the conjecture that appears in [7] to the case of multiplicative * -Lie-type maps:…”

“…In [4], the authors proved that a map Φ between two factor von Newmann algebras is a * -ring isomorphism if and only if Φ([X, Y ] * ) = [Φ(X), Φ(Y )] * . In [7], Ferreira and Costa extended these new products and defined two other types of applications, named multiplicative * -Lie n-map and multiplicative * -Jordan n-map and proved that, under certain conditions, an application between C * -algebras that is multiplicative * -Lie n-map and multiplicative * -Jordan n-map is a * -ring isomorphism. In the second paper of this series [6], Ferreira and Costa prove when a multiplicative * -Jordan n-map is a * -ring isomorphism.…”

“…With this picture in mind, in this article we will discuss when a multiplicative * -Lie n-map is a * -isomorphism and, just as it was done in [6], we provide an application on von Neumann algebras, factor von Neumann algebras and prime algebras. Let us define the following sequence of polynomials, as presented in [7]:…”

“…Note that p 2 * is the product introduced by Brešar and Fošner [2,15]. Then, using the nomenclature introduced in [7] we have a new class of maps (not necessarily additive): given two rings R and R ′ ,…”