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

TL;DR

This paper establishes a Kadec-Pelczyński dichotomy-type theorem for bounded sequences in the predual of JBW*-algebras, revealing structural decompositions involving orthogonal projections and weak compactness.

Contribution

It introduces a novel dichotomy theorem for preduals of JBW*-algebras, extending classical results to a non-commutative setting with specific decompositions.

Findings

Existence of subsequences with orthogonal projections

Decomposition into weakly compact and orthogonal parts

Structural insight into preduals of JBW*-algebras

Abstract

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 $(\phi_n)$ in the predual of a JBW$^*$-algebra $M$, there exist a subsequence $(\phi_{\tau(n)})$, and a sequence of mutually orthogonal projections $(p_n)$ in $M$ such that: [$(a)$] the set ${\phi_{\tau(n)} - \phi_{\tau(n)} P_{2} (p_n): n\in \mathbb{N}}$ is relatively weakly compact, $\phi_{\tau(n)}=\xi_n+\psi_n$, with $\xi_n := \phi_{\tau(n)} - \phi_{\tau(n)} P_{2} (p_n)$, and $\psi_n := \phi_{\tau(n)} P_{2} (p_n),$ {\rm(}$\xi_n Q(p_n)= 0$ and $\psi_n Q(p_n)^2 = \psi_n${\rm)}, for every $n$.