Bishop-Phelps-Bollobás property for positive operators when the domain is L∞

We prove that the class of positive operators from [Formula: see text] to [Formula: see text] has the Bishop-Phelps-Bollobás property for any positive measure [Formula: see text], whenever [Formula: see text] is a uniformly monotone Banach lattice with a weak unit. The same result also holds for the pair [Formula: see text] for any uniformly monotone Banach lattice [Formula: see text] Further we show that these results are optimal in case that [Formula: see text] is strictly monotone.

Operations that preserve integrability, and truncated Riesz spaces

For any real number {p\in[1,+\infty)}, we characterise the operations {\mathbb{R}^{I}\to\mathbb{R}} that preserve p-integrability, i.e., the operations under which, for every measure μ, the set {\mathcal{L}^{p}(\mu)} is closed. We investigate the infinitary variety of algebras whose operations are exactly such functions. It turns out that this variety coincides with the category of Dedekind σ-complete truncated Riesz spaces, where truncation is meant in the sense of R. N. Ball. We also prove that {\mathbb{R}} generates this variety. From this, we exhibit a concrete model of the free Dedekind σ-complete truncated Riesz spaces. Analogous results are obtained for operations that preserve p-integrability over finite measure spaces: the corresponding variety is shown to coincide with the much studied category of Dedekind σ-complete Riesz spaces with weak unit, {\mathbb{R}} is proved to generate this variety, and a concrete model of the free Dedekind σ-complete Riesz spaces with weak unit is exhibited.

On hull classes of ℓ-groups and covering classes of spaces

W denotes the category of archimedean ℓ-groups with designated weak unit and ℓ-homomorphisms that preserve the weak unit. Comp denotes the category of compact Hausdorff spaces with continuous maps. The Yosida functor is used to investigate the relationship between hull classes in W and covering classes in Comp. The central idea is that of a hull class whose hull operator preserves boundedness. We demonstrate how the Yosida functor may be used to identify hull classes in W and covering classes in Comp. In addition, we exhibit an array of order preserving bijections between certain families of hull classes and all covering classes, one of which was recently produced by Martínez. Lastly, we apply our results to answer a question of Knox and McGovern about the class of all feebly projectable ℓ-groups.

Functorial polar functions

W∞ denotes the category of archimedean ℓ-groups with designated weak unit and complete ℓ-homomorphisms that preserve the weak unit. CmpT2,∞ denotes the category of compact Hausdorff spaces with continuous skeletal maps. This work introduces the concept of a functorial polar function on W∞ and its dual a functorial covering function on CmpT2,∞.We demonstrate that functorial polar functions give rise to reflective hull classes in W ∞ and that functorial covering functions give rise to coreflective covering classes in CmpT 2,∞. We generate a variety of reflective and coreflecitve subcategories and prove that for any regular uncountable cardinal α, the class of α-projectable ℓ-groups is reflective in W ∞, and the class of α-disconnected compact Hausdorff spaces is coreflective in CmpT 2,∞. Lastly, the notion of a functorial polar function (resp. functorial covering function) is generalized to sublattices of polars (resp. sublattices of regular closed sets).

Existence of Invariant Weak Units in Banach Lattices: Countably Generated Left Amenable Semigroup of Operators

Let Σ be a countably generated left amenable semigroup and ﹛Tσ|σ ∈ Σ﹜ be a representation of Σ as a semigroup of positive linear operators on a weakly sequentially complete Banach lattice E with a weak unit e. It is assumed Tσ are uniformly bounded. It is shown that a necessary and sufficient condition for the existence of a weak unit invariant under ﹛Tσ | σ ∈ Σ﹜ is that inf σ∈Σ H(Tσe) > 0 for all nonzero H in the positive dual cone of E.