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, $$ \Phi(T)= UTU^* \; \text{for all} \;T\in B(H).$$