Small Riesz spaces

Abstract

Many facts in the theory of general Riesz spaces are easily verified by thinking in terms of spaces of functions. A proof via this insight is said to use representation theory. In recent years a growing number of authors has successfully been trying to bypass representation theorems, judging them to be extraneous. (See, for instance, [9,10].) In spite of the positive aspects of these efforts the following can be said. Firstly, avoiding representation theory does not always make the facts transparent. Reading the more cumbersome constructions and procedures inside the Riesz space itself one feels the need for a pictorial representation with functions, and one suspects the author himself of secret heretical thoughts. Secondly, the direct method leads to repeating constructions of the same nature over and over again.

Polar decomposition of order bounded disjointness preserving operators

“…It is the next result (Theorem 4.1 in [5]) that enables us to get around the nonconstructive extension argument used in Corollaries 3.1.19 and 3.1.20 of [13]. The previous theorem was proved for real Riesz spaces in [5] with an elementary proof, and the extension to the complex case is straightforward.…”

“…It is the next result (Theorem 4.1 in [5]) that enables us to get around the nonconstructive extension argument used in Corollaries 3.1.19 and 3.1.20 of [13]. The previous theorem was proved for real Riesz spaces in [5] with an elementary proof, and the extension to the complex case is straightforward. An elementary proof of the next proposition in the complex case follows immediately from its real version which is contained in Corollary 3 of [6] (for the case of orthomorphisms see Lemma 3.3 (ii) in [5]).…”

“…Thus we derive Theorem 2 above in a constructive way, underscoring once again that if the existence of a mathematical object is at stake (polar decomposition in this case) no Axiom of Choice is needed when uniqueness is assured. Amongst the results in [5], all of which are valid in Zermelo-Fraenkel set theory, we find Theorem 4.1, an extension theorem for orthomorphisms, which is one of the main ingredients of our approach here to polar decomposition theorems for order bounded disjointness preserving operators. Where the two theorems above prove a relationship between an order bounded disjointness preserving operator T : E → F and its absolute value |T |, we show a similar relation for certain operators S for which S(E) is somewhat related to |T | (E) and we study the related question of which operators S can be decomposed as a product W T , where W is an orthomorphism.…”

“…The previous theorem was proved for real Riesz spaces in [5] with an elementary proof, and the extension to the complex case is straightforward. An elementary proof of the next proposition in the complex case follows immediately from its real version which is contained in Corollary 3 of [6] (for the case of orthomorphisms see Lemma 3.3 (ii) in [5]). It can also be found as the first part of Theorem 3.3 in [2].…”

“…The paper [8] falls into that Zaanen program. An alternative approach to Zaanen's program was outlined in [5] and is based on the idea that sufficiently elementary results only involve countably many elements of the Riesz spaces under consideration. In this alternative way, one bypasses the very representation theorems that Grobler and Huijsmans wish to avoid, while one reasons in so-called small Riesz spaces only.…”