Abstract
In vector lattices, the concept of a projection band is a basic tool. We deal with projection bands in the more general setting of an Archimedean pre-Riesz space X. We relate them to projection bands in a vector lattice cover Y of X. If X is pervasive, then a projection band in X extends to a projection band in Y, whereas the restriction of a projection band B in Y is not a projection band in X, in general. We give conditions under which the restriction of B is a projection band in X. We introduce atoms and discrete elements in X and show that every atom is discrete. The converse implication is true, provided X is pervasive. In this setting, we link atoms in X to atoms in Y. If X contains an atom \(a>0\), we show that the principal band generated by a is a projection band. Using atoms in a finite dimensional Archimedean pre-Riesz space X, we establish that X is pervasive if and only if it is a vector lattice.
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Notes
There are several alternative definitions of order convergence of nets in partially ordered vector spaces. However, by [6, Theorem 4.4] an operator T is order continuous if and only if T is continuous with respect to any of these alternative notions.
We stress that in [2, Definition 1.42] the authors introduce this concept using a different term. They call an element \(a\in X_+\) for which \(0\leqslant x\leqslant a\) implies that \(x=\lambda a\) for some real \(\lambda \geqslant 0\) an extremal vector or a discrete vector of \(X_+\). However, a similar concept is well-known in the less general setting of vector lattices. Indeed, [20, § 3] gives a definition of an atom in a vector lattice which differs from our notion by the fact that the considered element need not be positive. However, by [24, III.13.1 a)] for every atom \(a\in Y\) it follows \(a>0\) or \(a<0\). Moreover, if \(a<0\) is an atom, so is \(-a\). These circumstances maybe clarify that, based on [2, Definition 1.42], we define atoms as positive elements. Proposition 26 justifies our choice in Archimedean pervasive pre-Riesz spaces.
For instance, in [16, Definition 26.1] the terms atom and discrete element are used in a reversed way with respect to our terminology.
Minkowski’s theorem states that in a finite-dimensional normed vector space every element of a compact convex set \(S\subseteq X\) is a convex combination of extreme points of S, see e.g. [19, p. 1].
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Kalauch, A., Malinowski, H. Projection bands and atoms in pervasive pre-Riesz spaces. Positivity 25, 177–203 (2021). https://doi.org/10.1007/s11117-020-00757-7
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DOI: https://doi.org/10.1007/s11117-020-00757-7
Keywords
- Order projection
- Projection band
- Band projection
- Principal band
- Atom
- Discrete element
- Extremal vector
- Pervasive
- Weakly pervasive
- Archimedean directed ordered vector space
- Pre-Riesz space
- Vector lattice cover