Commuting maps with the Mean Transform under Jordan product

Abstract

In this article, we give a complete characterization of the bijective maps which commute with the mean transform under Jordan product. The main result is the following : Let $H,K$ be two complex Hilbert spaces and $\Phi :B(H)\to B(K)$ be a bijective map, then $$\mathcal{M}(\Phi (A)\circ \Phi (B))=\Phi (\mathcal{M}(A\circ B))\text{for all}A,B\in B(H)$$ if and only if there exists a unitary or anti-unitary operator $U:H\to K$ such that, $$\Phi (T)=UT{U}^{\ast }\text{for all}T\in B(H).$$