Let $\mathfrak{ A}$ and $\mathfrak{ B}$ be JBW$^*$-algebras whose sets of unitaries are denoted by $\mathcal{ U}(\mathfrak{ A})$ and $\mathcal{ U}(\mathfrak{ B})$, respectively. We show that $\mathcal{ U}(\mathfrak{ A})$ is closed for Jordan products of operator commuting pairs inside itself. Assuming that $\mathfrak{ A}$ and $\mathfrak{ B}$ are JBW$^*$-algebras without direct summands of type $I_1$ or $I_2$, we prove that for each bicontinuous bijection $\Phi : \mathcal{ U}(\mathfrak{ A}) \rightarrow \mathcal{ U}(\mathfrak{ B})$ satisfying $\Phi (u \circ v) = \Phi (u)\circ \Phi (v),$ whenever $u$ and $v$ are operator commuting unitaries in $\mathfrak{ A}$, there exist a linear Jordan $^*$-isomorphism $\theta: \mathfrak{ A} \rightarrow \mathfrak{ B}$, a real linear mapping $\beta: \mathfrak{ A_{ sa}}\rightarrow Z(\mathfrak{ B}_{ sa})$, and an invertible central element $c \in \mathfrak{ B}_{ sa}$ such that $$ \Phi(e^{ i a}) = e^{ i \beta (a)}\circ e^{ i c\circ\theta(a)} = e^{ i \beta(a)} \circ \theta \left( e^{ i \theta^{ -1}( c )\circ a}\right),$$ for all $a\in \mathfrak{ A}_{ sa}$. The conclusion improves when $\mathfrak{ A}$ is a JBW$^*$-algebra factor not of type $I_2$.
Categories: math.OA math.FA