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

We prove a Kadec-Pelczyński dichotomy-type theorem for bounded sequences in the predual of a JBW*-algebra, showing that for each bounded sequence ( ϕ n ) in the predual of a JBW*-algebra M , there exist a subsequence ( ϕ τ ( n ) , and a sequence of mutually orthogonal projections ( p n ) in M such that: the set is relatively weakly compact ϕ τ ( n ) = ξ n + ψ n , with ξ n := ϕ τ ( n ) − ϕ τ ( n ) P 2 ( p n ), and ψ n := ϕ τ ( n ) P 2 ( p n ), ( ξ n Q ( p n ) = 0 and ψ n Q ( p n ) 2 = ψ n ) for every n .