Commuting maps with the Mean Transform under Jordan product

TL;DR

This paper characterizes bijective maps on operator algebras that commute with the mean transform under Jordan product, showing they are implemented by unitary or anti-unitary operators.

Contribution

It provides a complete characterization of maps commuting with the mean transform under Jordan product on operator algebras, identifying them as conjugations by unitary or anti-unitary operators.

Findings

Maps commuting with the mean transform are characterized as conjugations by unitary or anti-unitary operators.

The result applies to bijective maps on bounded operators between complex Hilbert spaces.

The characterization is complete and covers all such commuting maps.

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