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 $T:L\to M$ 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 $L\to M$, $T$ is normal, i.e., it preserves arbitrary suprema and infima;

4) for all $a\in M$,

$$a=sup\{x\in T(L):x\le a\}=inf\{x\in T(L):x\ge a\}.$$

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 $M\to K$.

The Riesz spaces $L_p(\mu )$( $1\le p<\mathrm{\infty }$) are Dedekind complete; so is $L_{\mathrm{\infty }}(\mu )$ 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 $l_{\mathrm{\infty }}$. 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 $T:L\to M$ is obvious.)

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