Vector lattice covers of ideals and bands in pre-Riesz spaces

Pre-Riesz spaces are ordered vector spaces which can be order densely embedded into vector lattices, their so-called vector lattice covers. Given a vector lattice cover $Y$ for a pre-Riesz space $X$, we address the question how to find vector lattice covers for subspaces of $X$, such as ideals and bands. We provide conditions such that for a directed ideal $I$ in $X$ its smallest extension ideal in $Y$ is a vector lattice cover. We show a criterion for bands in $X$ and their extension bands in $Y$ as well. Moreover, we state properties of ideals and bands in $X$ which are generated by sets, and of their extensions in $Y$

Positive-off-diagonal Operators on Ordered Normed Spaces and Maximum Principles for M-Operators

M-matrices are extensively employed in numerical analysis. These matrices can be generalized by corresponding operators on a partially ordered normed space. We extend results which are well-known for M-matrices to this more general setting. We investigate two different notions of an M-operator, where we focus on two questions: 1. For which types of partially ordered normed spaces do the both notions coincide? This leads to the study of positive-off-diagonal operators. 2. Which conditions on an M-operator ensure that its (positive) inverse satisfies certain maximum principles? We deal with generalizations of the "maximum principle for inverse column entries".M-Matrizen werden in der numerischen Mathematik vielfältig angewandt. Eine Verallgemeinerung dieser Matrizen sind entsprechende Operatoren auf halbgeordneten normierten Räumen. Bekannte Aussagen aus der Theorie der M-Matrizen werden auf diese Situation übertragen. Für zwei verschiedene Typen von M-Operatoren werden die folgenden Fragen behandelt: 1. Für welche geordneten normierten Räume sind die beiden Typen gleich? Dies führt zur Untersuchung außerdiagonal-positiver Operatoren. 2. Welche Bedingungen an einen M-Operator sichern, dass seine (positive) Inverse gewissen Maximumprinzipien genügt? Es werden Verallgemeinerungen des "Maximumprinzips für inverse Spalteneinträge" angegeben und untersucht

Order theoretical structures in atomic JBW-algebras: disjointness, bands, and centres

Every atomic JBW-algebra is known to be a direct sum of JBW-algebra factors of type I. Extending Kadison's anti-lattice theorem, we show that each of these factors is a disjointness free anti-lattice. We characterise disjointness, bands, and disjointness preserving bijections with disjointness preserving inverses in direct sums of disjointness free anti-lattices and, therefore, in atomic JBW-algebras. We show that in unital JB-algebras the algebraic centre and the order theoretical centre are isomorphic. Moreover, the order theoretical centre is a Riesz space of multiplication operators. A survey of JBW-algebra factors of type I is included

A Hahn-Jordan decomposition and Riesz-Frechet representation theorem in Riesz spaces

We give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of [{{\sc B. A. Watson}, {An And\^o-Douglas type theorem in Riesz spaces with a conditional expectation,} {\em Positivity,} {\bf 13} (2009), 543 - 558}] and a Riesz-Frechet representation theorem for the $T$-strong dual, where $T$ is a Riesz space conditional expectation operator. The result of Watson was formulated specifically to assist in the proof of the existence of Riesz space conditional expectation operators with given range space, i.e., a result of And\^{o}-Douglas type. This was needed in the study of Markov processes and martingale theory in Riesz spaces. In the current work, our interest is a Riesz-Frechet representation theorem, for which another variant of the Hahn-Jordan decomposition is required

Inverses of disjointness preserving operators in finite dimensional pre-Riesz spaces

If ℝn is partially ordered by a generating closed cone K; then (ℝn;K) is a pre-Riesz space. We show for a disjointness preserving bijection T on (ℝn;K) that the inverse of T is also disjointness preserving. We prove that for T there is k ∈ P(b) such that Tk is band preserving, where b is the number of bands in (ℝn;K); and P(b) the set of orders of permutations on b symbols.Mathematics Subject Classification (2010): 47B60.Keywords: Band, band preserving operator, disjointness preserving operator, d-isomor-phism, nite dimensional, generating closed cone, pre-Riesz space

On maximum principles for M-operators

AbstractFor M-matrices a condition to satisfy the “maximum principle for inverse column entries” is known. We generalize this result (concerning a more general maximum principle) for M-operators on Rn, ordered by some cone, as well as, to a certain extent, for M-operators on infinite-dimensional ordered normed spaces