Let $A$ and $B$ be unital $C$*-algebras and let $T\colon A\to B$ be a bounded linear bijection that preserves commuting elements, i.e., $ab=ba$ implies $TaTb=TbTa$. Does $T^{**}$ then also preserve commuting elements?
An approach to answer this question would be to weak*-approximate commuting elements in $A^{**}$ by commuting elements of $A$, but by Commuting nets for commuting projections this already seems to fail for projections.
