In this paper, we characterize when, for any infinite cardinal α , the Fremlin tensor product of two Archimedean Riesz spaces (see Fremlin in Am J Math 94:777–798, 1972) is Dedekind α -complete. We also provide an example of an ideal I in an Archimedean Riesz space E such that the Fremlin tensor product of I with itself is not an ideal in the Fremlin tensor product of E with itself.

We show that the Carathéodory space of place functions on the free product of two Boolean algebras is Riesz isomorphic with Fremlin’s Archimedean Riesz space tensor product of their respective Carathéodory spaces of place functions. We provide a solution to Fremlin’s problem 315Y(f) [Measure Theory, Torres Fremlin, Colchester, 2004] concerning completeness in the free product of Boolean algebras by applying our results on the Archimedean Riesz space tensor product to Carathéodory spaces of place functions.

We prove that polynomial valuations on vector lattices correspond to orthosymmetric multilinear maps. As a consequence we obtain a concise proof of the correspondence between orthosymmetry and orthogonal additivity.

We prove that polynomial valuations on vector lattices correspond to orthosymmetric multilinear maps. As a consequence we obtain a concise proof of the equivalence of orthosymmetry and orthogonal additivity.

We derive formulas for characterizing bounded orthogonally additive polynomials in two ways. Firstly, we prove that certain formulas for orthogonally additive polynomials derived in Kusraeva (Vladikavkaz Math J 16(4):49–53, 2014) actually characterize them. Secondly, by employing complexifications of the unique symmetric multilinear maps associated with orthogonally additive polynomials, we derive new characterizing formulas.

Among other results, we provide a formula for the order continuous component of a positive s-homogeneous polynomial on a vector lattice with Dedekind complete range.

We prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean (real or complex) vector lattice, from which the Cauchy-Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean (real or complex) $f$-algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove the H\"older inequality for weighted geometric mean closed Archimedean (real or complex) $\Phi$-algebras, improving results by Boulabiar and Toumi. As a consequence, the Minkowski inequality for weighted geometric mean closed Archimedean (real or complex) $\Phi$-algebras is obtained.

This workshop focused on recent developments in the area of Banach lattices. The goal of the workshop was to bring together leading experts and active young researchers to discuss the current and future directions of these developments and to identify potential applications and the main open problems. We planned to understand the ”big picture” of connections between these developments and other areas of Functional Analysis. The workshop focused on the following topics:

AbstractIn this short note, we correct and reformulate Theorem 3.1 in the paper published in Proceedings of the Edinburgh Mathematical Society58(3) (2015), 617–629.

In this short note, we correct and reformulate Theorem 3.1 in the paper published in Proceedings of the Edinburgh Mathematical Society 58(3) (2015), 617-629.

Frolik's theorem says that a homeomorphism from a certain kind of topological space to itself decomposes the space into the clopen set of fixed points together with three clopen sets, each of whose images is disjoint from the original set. Stone's theorem translates this result to a corresponding theorem about the Riesz space of continuous functions on the topological space. We prove a theorem analogous to that for Riesz spaces in the much more general setting of (possibly noncommnutative) lattice-ordered groups and group-endomorphisms. The groups to which our result applies satisfy a weak condition, introduced by Abramovich and Kitover, on the polars; the images of our endomorphisms have a kind of order-density on their polars; the double polars of the images are cardinal summands; and the endomorphisms themselves are disjointness-preserving in both directions. We explain how to extend our result to larger groups to which it does not apply, and, to give additional insight, we provide many examples.

We construct the tensor product for f-algebras, including proving a universal property for it, and investigate how it preserves algebraic properties of the factors.

We study completions of Archimedean vector lattices relative to any nonempty set of positively homogeneous functions on finite-dimensional real vector spaces. Examples of such completions include square mean closed and geometric mean closed vector lattices, amongst others. These functional completions also lead to a universal definition of the complexification of any Archimedean vector lattice and a theory of tensor products and powers of complex vector lattices in a companion paper.

We show that the Fremlin tensor product C(X)⊗¯C(Y) is not square mean complete when X and Y are uncountable metrizable compact spaces. This motivates the definition of complexification of Archimedean vector lattices, the Fremlin tensor product of Archimedean complex vector lattices, and a theory of powers of Archimedean complex vector lattices.

In this paper we use the norm of bounded variation to study multilinear operators and polynomials on Banach lattices. As a result, we obtain when all continuous multilinear operators and polynomials on Banach lattices are regular. We also provide new abstract M- and abstract L-spaces of multilinear operators and polynomials and generalize all the results by Grecu and Ryan, from Banach lattices with an unconditional basis to all Banach lattices.

Let E be a Banach space with 1-unconditional basis. Denote by Delta((circle times) over cap (n,pi) E) (resp. Delta((circle times) over cap (n,s,pi) E)) the main diagonal space of the n-fold full (resp. symmetric) projective Banach space tensor product, and denote by Delta((circle times) over cap (n,vertical bar pi vertical bar) E) (resp. Delta((circle times) over cap (n,s,vertical bar pi vertical bar) E)) the main diagonal space of the n-fold full (resp. symmetric) projective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic, and in addition, that they are isometrically lattice isomorphic to E-[n], the completion of the n-concavification of E. Using these isometries, we also show that the norm of any (vector valued) continuous orthogonally additive homogeneous polynomial on E equals the norm of its associated symmetric linear operator.

We introduce tensor products in the category of lattice seminormed spaces. We show that the reasonable cross vector seminorms on the complexification of a lattice seminormed space are the same as the admissible vector seminorms. We then specialize these results to complexifications of Archimedean Riesz spaces.

We investigate the relationship between the diagonal of the Fremlin projective tensor product of a Banach lattice E with itself and the 2-concavification of E.

This paper is the first systematic study of homogeneous polynomials on Banach lattices. A variety of new Banach spaces and Banach lattices of multilinear maps, homogeneous polynomials, and operators are introduced. The main technique is to employ positive tensor products and quotients of positive tensor products. Our theorems generalize the results on orthogonally additive polynomials by Benyamini, Lassalle, and Llavona (2006) in [4], the results by Grecu and Ryan (2005) in [14], and the results by Sundaresan (1991) in [23].