Riesz seminorms with Fatou properties

Abstract

A seminormed Riesz space L ρ {L_\rho } satisfies the σ \sigma -Fatou property (resp. the Fatou property) if θ ≤ u n ↑ u \theta \leq {u_n} \uparrow u in L L (resp. θ ≤ u α ↑ u in L \theta \leq {u_\alpha } \uparrow u\;{\text {in}}\;L ) implies ρ ( u n ) ↑ ρ ( u ) \rho ({u_n}) \uparrow \rho (u) (resp. ρ ( u α ) ↑ ρ ( u ) \rho ({u_\alpha }) \uparrow \rho (u) ). The following results are proved: (i) A normed Riesz space L ρ {L_\rho } satisfies the σ \sigma -Fatou property if, and only if, its norm completion does and L ρ {L_\rho } has ( A , 0 ) ({\mathbf {A}},0) . (ii) The quotient space L ρ / I ρ {L_\rho }/{I_\rho } has the Fatou property if L ρ {L_\rho } is Archimedean with the Fatou property. ( I ρ = { u ε L : ρ ( u ) = 0 } . ) ({I_\rho } = \{ u\varepsilon L:\rho (u) = 0\} .) (iii) If L ρ {L_\rho } is almost σ \sigma -Dedekind complete with the σ \sigma -Fatou property, then L ρ / I ρ {L_\rho }/{I_\rho } has the σ \sigma -Fatou property. A counterexample shows that (iii) may be false for Archimedean Riesz spaces.