A Kadec-Pelczyński dichotomy-type theorem for preduals of JBW*-algebras

Abstract

We prove a Kadec-Pelczyński 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 ${\varphi }_{\tau (n)}-{\varphi }_{\tau (n)}{P}_2(p_n):n\in \mathbb{N}$ is relatively weakly compact, ${\varphi }_{\tau (n)}={\xi }_n+{\psi }_n$, with ${\xi }_n:={\varphi }_{\tau (n)}-{\varphi }_{\tau (n)}{P}_2(p_n)$, and ${\psi }_n:={\varphi }_{\tau (n)}{P}_2(p_n),$ {\rm(}${\xi }_{n}Q(p_n)=0$ and ${\psi }_{n}Q(p_n{)}^2={\psi }_n${\rm)}, for every $n$.