Quadratic & additive mappings on operator commuting elements in JBW*-algebras

Abstract

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 $I_2$.