Dedekind completion

of a Riesz space

A Riesz space is called Dedekind complete if every non-empty subset that is bounded from below (respectively, above) has an infimum (respectively, supremum). A Dedekind-complete Riesz space is automatically Archimedean. Hence, so are its Riesz subspaces.

Given an Archimedean Riesz space $ L $, a Dedekind completion of $ L $ is a pair $ ( M,T ) $ where $ M $ is a Riesz space and $ T : L \rightarrow M $ is a mapping such that

1) $ M $ is Dedekind complete;

2) $ T $ is a Riesz isomorphism of $ L $ onto a Riesz subspace $ T ( L ) $ of $ M $;

3) as a mapping $ L \rightarrow M $, $ T $ is normal, i.e., it preserves arbitrary suprema and infima;

4) for all $ a \in M $,

$$ a = \sup \left \{ {x \in T ( L ) } : {x \leq a } \right \} = \inf \left \{ {x \in T ( L ) } : {x \geq a } \right \} . $$

Every Archimedean Riesz space $ L $ has a Dedekind completion, whose underlying partially ordered set can be obtained from the MacNeille completion (cf. Completion, MacNeille (of a partially ordered set)) of $ L $ by removing its largest and smallest elements. The Dedekind completion is unique in the following sense. If $ ( M _ {1} ,T _ {1} ) $ and $ ( M _ {2} ,T _ {2} ) $ are Dedekind completions of $ L $, then there exists a unique Riesz isomorphism $ S $ of $ M _ {1} $ onto $ M _ {2} $ with $ T _ {2} = S \circ T _ {1} $. More generally, if $ ( M,T ) $ is a Dedekind completion of $ L $, then every normal Riesz homomorphism of $ L $ into any Dedekind-complete Riesz space $ K $ can uniquely be extended to a normal Riesz homomorphism $ M \rightarrow K $.

The Riesz spaces $ L _ {p} ( \mu ) $( $ 1 \leq p < \infty $) are Dedekind complete; so is $ L _ \infty ( \mu ) $ if $ \mu $ is $ \sigma $- finite. The space $ C ( X ) $( $ X $ a compact Hausdorff space) is Dedekind complete if and only if $ X $ is extremally disconnected (cf. Extremally-disconnected space). There are few non-trivial instances of Riesz spaces whose Dedekind completions are to some extent "understood" . The Dedekind completion of the space $ c $ of all converging sequences is $ l _ \infty $. That of $ C ( X ) $ is the quotient $ B ( X ) /N $, where $ B ( X ) $ is the space of all bounded Borel functions and $ N $ is the ideal of all functions that vanish off meager sets (cf. Category of a set). (In either case, the mapping $ T : L \rightarrow M $ is obvious.)

References

[a1]	E. de Jonge, A. van Rooij, "Introduction to Riesz spaces" , Tracts , 8 , Math. Centre, Amsterdam (1977)
[a2]	W. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1974)
[a3]	B.Z. Vulikh, "Introduction to the theory of partially ordered spaces" , Wolters–Noordhoff (1967) (In Russian)