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

TL;DR

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

Contribution

It provides a characterization of bijective unital multiplicative *-Jordan-type maps on $C^*$-algebras, especially in factor von Neumann algebras, establishing their equivalence to *-ring isomorphisms.

Findings

Bijective unital multiplicative *-Jordan-type maps are *-ring isomorphisms in factor von Neumann algebras.

Characterization of *-Jordan-type maps on general $C^*$-algebras.

Structural insight into the nature of *-maps preserving Jordan products.

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 projections in $\mathfrak{A}$. In this paper we study the characterization of multiplicative $*$-Jordan-type maps. In particular, if $\mathcal{M}$ is a factor von Neumann algebra then every bijective unital multiplicative $*$-Jordan-type maps are $*$-ring isomorphisms.