Convergence and boundedness criteria in the bounded weak-star function space

Derive efficient criteria for uo convergence and order convergence in $C_{ph}(E^{*},\mathrm{bw}^{*})$, and characterize its order-bounded and dominable subsets.

Background

The paper identifies Cph(E∗,bw∗)C_{ph}(E^{*},\mathrm{bw}^{*}) with the free AM-space representation and proves that it is a regular sublattice of C(E∗,bw∗)C(E^{*},\mathrm{bw}^{*}). Although uo and order convergence can be inherited from the ambient continuous-function lattice in principle, the paper states that more efficient criteria exploiting positive homogeneity and the bounded weak-star topology are not provided.

References

Find the most efficient criteria for uo and order convergence in $C_{ph}\left(E{},\mathrm{bw}{}\right)$, as well as descriptions of order bounded and dominable sets.

— Variants of order semicontinuity in Banach lattices  (2609.03070 - Bilokopytov, 2 Sep 2026) in Question following Corollary \ref{regu2}, Section 8, “Semicontinuity-like properties in free AM-spaces”