A Kadec-Pelczy\'nski dichotomy-type theorem for preduals of JBW*-algebras

We prove a Kadec-Pelczy'nski dichotomy-type theorem for bounded sequences in the predual of a JBW*-algebra, showing that for each bounded sequence $({\varphi }_n)$ in the predual of a JBW${}^{\ast }$-algebra $M$, there exist a subsequence $({\varphi }_{\tau (n)})$, and a sequence of mutually orthogonal projections $(p_n)$ in $M$ such that: [$(a)$] the set ${\phi_{\tau(n)} - \phi_{\tau(n)} P_{2} (p_n): n\in $\mathbb{N}$}$isrelativelyweaklycompact,$\phi_{\tau(n)}=\xi_n+\psi_n$,with$\xi_n := \phi_{\tau(n)} - \phi_{\tau(n)} P_{2} (p_n)$,and$\psi_n := \phi_{\tau(n)} P_{2} (p_n),$($\xi_n Q(p_n)= 0$and$\psi_n Q(p_n)^2 = \psi_n$),forevery$n$.