Riesz seminorms with Fatou properties

Abstract:

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