Disjointness preserving C0-semigroups and local operators on ordered Banach spaces

We generalize results concerning ${\mathrm{C}}_0$-semigroups on Banach lattices to a setting of ordered Banach spaces. We prove that the generator of a disjointness preserving ${\mathrm{C}}_0$-semigroup is local. Some basic properties of local operators are also given. We investigate cases where local operators generate local ${\mathrm{C}}_0$-semigroups, by using Taylor series or Yosida approximations. As norms we consider regular norms and show that bands are closed with respect to such norms. Our proofs rely on the theory of embedding pre-Riesz spaces in vector lattices and on corresponding extensions of regular norms.     

Riesz completions,functional representations,and anti-lattices

We show that the Riesz completion of an Archimedean partially ordered vector space $X$ with unit can be represented as a norm dense Riesz subspace of the smallest functional representation of $X.$ This yields a convenient way to find the Riesz completion. To illustrate the method, the Riesz completions of spaces ordered by Lorentz cones, cones of symmetric positive semi-definite matrices, and polyhedral cones are determined. We use the representation to analyse the existence of non-trivial disjoint elements and link the absence of such elements to the notion of anti-lattice. One of the results is a geometric condition on the dual cone of a finite dimensional partially ordered vector space $X$ that ensures that $X$ is an anti-lattice.     

Operators in pre-Riesz spaces: moduli and homomorphisms

We focus on two topics that are related to moduli of elements in partially ordered vector spaces. First, we relate operators that preserve moduli to generalized notions of lattice homomorphisms, such as Riesz homomorphisms, Riesz* homomorphisms, and positive disjointness preserving operators. We also consider complete Riesz homomorphisms, which generalize order continuous lattice homomorphisms. Second, we characterize elements with a modulus by means of disjoint elements and apply this result to obtain moduli of functionals and operators in various settings. On spaces of continuous functions, we identify those differences of Riesz* homomorphisms that have a modulus. Many of our results for pre-Riesz spaces of continuous functions lead to results on order unit spaces, where the functional representation is used.

     

Directed ideals in partially ordered vector spaces

For a linear subspace I$I$ of a Riesz space there are various well-known properties that are equivalent to I$I$ being an ideal, such as I$I$ is a full Riesz subspace, I$I$ is a solid subspace, I$I$ is a Riesz subspace and the kernel of a positive linear map, I$I$ is the kernel of a Riesz homomorphism. Generalizations of these properties to partially ordered vector spaces are considered and their relations are investigated. It is shown that for directed subspaces all these generalizations are equivalent, just as in the case of Riesz spaces.     

Disjointness in Partially Ordered Vector Spaces

A notion of disjointness in arbitrary partially ordered vector spaces is introduced by calling two elements x and y disjoint if the set of all upper bounds of x + y and −x − y equals the set of all upper bounds of x − y and −x + y. Several elementary properties are easily observed. The question whether the disjoint complement of a subset is a linear subspace appears to be more difficult. It is shown that in directed Archimedean spaces disjoint complements are always subspaces. The proof relies on theory on order dense embedding in vector lattices. In a non-Archimedean directed space even the disjoint complement of a singleton may fail to be a subspace. According notions of disjointness preserving operator, band, and band preserving operator are defined and some of their basic properties are studied.     

Pervasive and weakly pervasive pre-Riesz spaces

Pervasive pre-Riesz spaces are defined by means of vector lattice covers. To avoid the computation of a vector lattice cover, we give two distinct intrinsic characterizations of pervasive pre-Riesz spaces. We introduce weakly pervasive pre-Riesz spaces and observe that this property can be easily checked in examples. We relate weakly pervasive pre-Riesz spaces to pre-Riesz spaces with the Riesz decomposition property.     

Tensor products of Archimedean partially ordered vector spaces

We study the tensor product of two directed Archimedean partially ordered vector spaces X and Y by means of Riesz completions. With the aid of the Fremlin tensor product of the Riesz completions of X and Y we show that the projective cone in X ? Y is contained in an Archimedean cone. The smallest Archimedean cone containing the projective cone satisfies an appropriate universal mapping property.     

On a certain Maximum Principle for Positive Operators in an Ordered Normed Space

The equation for a positive linear continuous operator is considered in an ordered normed space , where the cone is assumed to be closed and having a nonempty interior. Then the dual cone of K possesses a base . Generalizing the well known maximum principle for positive matrices an operator A is said to satisfy the maximum principle, if for any there exists a positive linear continuous functional which is both, maximal on the element Ax, i.e. , and positive on the element x, i.e. 0$$ " align="middle" border="0"> . This property is studied and characterized both analytically by some extreme point condition and geometrically by means of the behaviour under A of the faces of the cone K. It is shown that the conditions which have been obtained for finite dimensional spaces in earlier relevant papers are special cases of conditions presented in this paper. The maximum pinciple is proved for simple operators in the spaces and c.