Commuting maps with the Mean Transform under Jordan product

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 $Φ:B(H) \to B(K)$ be a bijective map, then $\mathcal {M}(Φ(A)\circΦ(B))=Φ(\mathcal{M}(A\circ B)) \;\; \text{for all}\;\; A, B \in B(H)$ if and only if there exists a unitary or anti-unitary operator $U:H\to K$ such that, $Φ(T)= UTU^* \; \text{for all} \;T\in B(H).$

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