arXiv subjects
Publications and source records attributed to Francisco J. FernÁndez-Polo.
We prove that every (not necessarily linear nor continuous) 2-local triple homomorphism from a JBW$^*$-triple into a JB$^*$-triple is linear and a triple homomorphism. Consequently, every 2-local triple homomorphism from a von Neumann algebra (respectively, from a JBW$^*$-algebra) into a C$^*$-algebra (respectively, into a JB$^*$-algebra) is linear and a triple homomorphism.
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: [$(a)$] the set ${ϕ_{τ(n)} - ϕ_{τ(n)} P_{2} (p_n): n\in \mathbb{N}}$ 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 $ψ_n := ϕ_{τ(n)} P_{2} (p_n),$ {\rm(}$ξ_n Q(p_n)= 0$ and $ψ_n Q(p_n)^2 = ψ_n${\rm)}, for every $n$.