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

Abstract

Let $\mathfrak{A}$ and $\mathfrak{B}$ be JBW${}^{\ast }$-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${}^{\ast }$-algebras without direct summands of type $I_1$ or $I_2$, we prove that for each bicontinuous bijection $\hat{I}¦:\mathcal{U}(\mathfrak{A})\to \mathcal{U}(\mathfrak{B})$ satisfying $\hat{I}¦(u\circ v)=\hat{I}¦(u)\circ \hat{I}¦(v),$ whenever $u$ and $v$ are operator commuting unitaries in $\mathfrak{A}$, there exist a linear Jordan ${}^{\ast }$-isomorphism $\hat{I}¸:\mathfrak{A}\to \mathfrak{B}$, a real linear mapping ${\hat{I}}^2:{\mathfrak{A}}_{\mathfrak{sa}}\to Z({\mathfrak{B}}_{sa})$, and an invertible central element $c\in {\mathfrak{B}}_{sa}$ such that $$\hat{I}¦(e^{ia})=e^{i{\hat{I}}^2(a)}\circ e^{ic\circ \hat{I}¸(a)}=e^{i{\hat{I}}^2(a)}\circ \hat{I}¸\left(e^{i\hat{I}{¸}^{-1}(c)\circ a}\right),$$ for all $a\in {\mathfrak{A}}_{sa}$. The conclusion improves when $\mathfrak{A}$ is a JBW${}^{\ast }$-algebra factor not of type $I_2$.