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

TL;DR

This paper characterizes multiplicative $*$-Lie-type maps on $C^*$-algebras, showing that in factor von Neumann algebras such maps are necessarily $*$-isomorphisms, thus revealing their structural nature.

Contribution

It provides a characterization of multiplicative $*$-Lie-type maps on $C^*$-algebras, especially demonstrating their equivalence to $*$-isomorphisms in factor von Neumann algebras.

Findings

Every bijective unital multiplicative $*$-Lie-type map on a factor von Neumann algebra is a $*$-isomorphism.

The paper characterizes these maps in terms of algebraic structure preservation.

It extends the understanding of $*$-Lie-type maps in the context of operator algebras.

Abstract

Let $\mathfrak{A}$ and $\mathfrak{A}'$ be two $C^*$-algebras with identities $I_{\mathfrak{A}}$ and $I_{\mathfrak{A}'}$, respectively, and $P_1$ and $P_2 = I_{\mathfrak{A}} - P_1$ nontrivial symmetric projections in $\mathfrak{A}$. In this paper we study the characterization of multiplicative $*$-Lie-type maps. In particular, if $\mathcal{M}$ is a factor von Neumann algebra then every complex scalar multiplication bijective unital multiplicative $*$-Lie-type map is $*$-isomorphism.