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

الأصل · EN

We prove a Kadec-Pelczyński dichotomy-type theorem for bounded sequences in the predual of a JBW*-algebra, showing that for each bounded sequence (ϕₙ) in the predual of a JBW*-algebra M, there exist a subsequence (ϕτ₍ₙ₎), and a sequence of mutually orthogonal projections (pₙ) in M such that: [(a)] the set ϕτ₍ₙ₎ - ϕτ₍ₙ₎ P₂ (pₙ): n∈ N is relatively weakly compact, ϕτ₍ₙ₎=ξₙ+ψₙ, with ξₙ:= ϕτ₍ₙ₎ - ϕτ₍ₙ₎ P₂ (pₙ), and ψₙ:= ϕτ₍ₙ₎ P₂ (pₙ), (ξₙ Q(pₙ)= 0 and ψₙ Q(pₙ)² = ψₙ), for every n.