Preservers of operator commutativity

Let $M$ and $J$ be JBW^⁎-algebras admitting no central summands of type $I_1$ and $I_2$, and let $\Phi :M\to J$ be a linear bijection preserving operator commutativity in both directions, that is,$[x,M,y]=0\iff \left[\Phi \right(x),J,\Phi (y\left)\right]=0,$ for all $x,y\in M$, where the associator of three elements $a,b,c$ in $M$ is defined by $[a,b,c]:=(a\circ b)\circ c-(c\circ b)\circ a$. We prove that under these conditions there exist a unique invertible central element $z_0$ in $J$, a unique Jordan isomorphism $J:M\to J$, and a unique linear mapping β from $M$ to the centre of $J$ satisfying$\Phi \left(x\right)=z_0\circ J\left(x\right)+\beta \left(x\right),$ for all $x\in M$. Furthermore, if Φ is a symmetric mapping (i.e., $\Phi \left(x^{⁎}\right)=\Phi {\left(x\right)}^{⁎}$ for all $x\in M$), the element $z_0$ is self-adjoint, J is a Jordan ^⁎-isomorphism, and β is a ^⁎-symmetric mapping too.

In case that $J$ is a JBW^⁎-algebra admitting no central summands of type $I_1$, we also address the problem of describing the form of all symmetric bilinear mappings $B:J\times J\to J$ whose trace is associating (i.e., $\left[B\right(a,a),b,a]=0$, for all $a,b\in J$) providing a complete solution to it. We also determine the form of all associating linear maps on $J$.