First we give a counterexample showing that recent results on separate order continuity of Arens extensions of multilinear operators cannot be improved to get separate order continuity on the product of the whole of the biduals. Then we establish conditions on the operators and/or on the underlying Riesz spaces/Banach lattices so that the extensions are order continuous on the product of the whole biduals. We also prove that all Arens extensions of any regular multilinear operator are order continuous in at least one variable and we study when Arens extensions of regular homogeneous polynomials on a Banach lattice $E$ are order continuous on $E^{**}$.

We show that for the case of uniformly convex Banach spaces, the conditions of Brøndsted’s fixed point theorem can be relaxed.

All non-negative, continuous, $\text{SL}(n)$, and translation invariant valuations on the space of super-coercive, convex functions on $\mathbb{R}^{n}$ are classified. Furthermore, using the invariance of the function space under the Legendre transform, a classification of non-negative, continuous, $\text{SL}(n)$, and dually translation invariant valuations is obtained. In both cases, different functional analogs of the Euler characteristic, volume, and polar volume are characterized.

We prove that Krivine’s Function Calculus is compatible with integration. Let $(\unicode[STIX]{x1D6FA},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$ be a finite measure space, $X$ a Banach lattice, $\mathbf{x}\in X^{n}$, and $f:\mathbb{R}^{n}\times \unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$ a function such that $f(\cdot ,\unicode[STIX]{x1D714})$ is continuous and positively homogeneous for every $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}$, and $f(\mathbf{s},\cdot )$ is integrable for every $\mathbf{s}\in \mathbb{R}^{n}$. Put $F(\mathbf{s})=\int f(\mathbf{s},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$ and define $F(\mathbf{x})$ and $f(\mathbf{x},\unicode[STIX]{x1D714})$ via Krivine’s Function Calculus. We prove that under certain natural assumptions $F(\mathbf{x})=\int f(\mathbf{x},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$, where the right hand side is a Bochner integral.

Dimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like; for instance, it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In an appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs.

Following ideas used by Drewnowski and Wilansky we prove that if $I$ is an infinite dimensional and infinite codimensional closed ideal in a complete metrizable locally solid Riesz space and $I$ does not contain any order copy of ${{\mathbb{R}}^{\mathbb{N}}}$ then there exists a closed, separable, discrete Riesz subspace $G$ such that the topology induced on $G$ is Lebesgue, $I\,\bigcap \,G\,=\,\left\{ 0 \right\}$, and $I\,+\,G$ is not closed.

It is proved that if a positive operator $S\,:\,E\,\to \,F$ is $\text{AM}$-compact whenever its adjoint ${{S}^{'}}:{{F}^{'}}\to {{E}^{'}}$ is $\text{AM}$-compact, then either the norm of $\text{F}$ is order continuous or $E\prime $ is discrete.

Given a metrizable locally convex-solid Riesz space of measurable functions we provide a procedure to construct a minimal Fréchet (function) lattice containing it, called its Fatou completion. As an application, we obtain that the Fatou completion of the space L1(ν) of integrable functions with respect to a Fréchet-space-valued measure ν is the space L1w(ν) of scalarly ν-integrable functions. Further consequences are also given.

We study the duality problem for order weakly compact operators by giving sufficient and necessary conditions under which the order weak compactness of an operator implies the order weak compactness of its adjoint and conversely.

We prove the converse of a theorem of Zaanen about the duality problem of positive $\text{AM}$-compact operators.

Let A be a uniformly complete vector sublattice of an Archimedean semiprime f-algebra B and p ∈ {1, 2,…}. It is shown that the set ΠBp (A) = {f1 … fp: fk ∈ A, k = 1, …, p } is a uniformly complete vector sublattice of B. Moreover, if A is provided with an almost f-algebra multiplication * then there exists a positive operator Tp, from ΠBp(A) into A such that fi *…* fp = Tp(f1 …fp) for all f1…fp ∈ A.

As application, being given a uniformly complete almost f-algebra (A, *) and a natural number p ≧ 3, the set Π*p(A) = {f1 *… *fp: fk ∈ A, k = 1…p} is a uniformly complete semiprime f-algebra under the ordering and the multiplication inherited from A.

We give a characterization of nuclear Fréchet lattices in terms of lattice properties of the seminorms. Indeed, we prove that a Fréchet lattice is nuclear if and only if it is both an AL- and an AM-space.

We consider the space L1 (ν, X) of all real functions that are integrable with respect to a measure v with values in a real Fréchet space X. We study L-weak compactness in this space. We consider the problem of the relationship between the existence of copies of l∞ in the space of all linear continuous operators from a complete DF-space Y to a Fréchet lattice E with the Lebesgue property and the coincidence of this space with some ideal of compact operators. We give sufficient conditions on the measure ν and the space X that imply that L1 (ν, X) has the Dunford-Pettis property. Applications of these results to Fréchet AL-spaces and Köthe sequence spaces are also given.

As a consequence of general principles, we add to the array of ‘hulls’ in the category Arch (of archimedean ℓ-groups with ℓ-homomorphisms) and in its non-full subcategory W (whose objects have distinguished weak order unit, whose morphisms preserve the unit). The following discussion refers to either Arch or W. Let α be an infinite cardinal number or ∞, let Homα; denote the class of α-complete homomorphisms, and let R be a full epireflective subcategory with reflections denoted rG: G → rG. Then for each G, there is rαG ∈ Homα (G, R) such that for each ϕ ∈ Homα (G, R), there is unique with . Moreover if every rG is an essential embedding, then, for every α and every G, rαG = rG, and every Homα. If and R consists of all epicomplete objects, then every Homw1. For α = ∞, and for any R, every Hom∞.

A vector lattice F is said to be Dedekind α-complete, where α is a cardinal number, provided that each non-empty order bounded subset D of F satisfying card(D) ≤ α has a supremum. Several characterizations of this property are presented here.

Let {Pi} be a sequence of real (Laurent) polynomials each of which has no negative coefficients, and suppose that f is a real polynomial. Consider the problem of deciding whether

for all integers k, there exists Nsuch that the product of polynomials

(*) Pk+1. Pk+2.....Pk+N·ƒ has no negative coefficients.

It is shown that if L is a Dedekind complete Riesz space equipped with a locally solid topology T defined by strongly (A, 0) Riesz pseudonorms, then order intervals of L are T-complete. This is an extension of a well known theorem of Nakano. The second part of the paper gives a necessary and sufficient condition for topological completeness of order intervals in a Dedekind σ-complete Riesz space which has a weak order unit and which is equipped with a locally solid σ-Fatou topology.