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Let $\ell_{1,0}(S)=(c_{0,0}(S),\Vert\cdot\Vert_1)$ be a space of functions on a set $S$ with finite support, endowed with $\ell_1$ norm. Could you answer the at least one of the following questions

1) which subspaces in $X$ are complementable?

2) which subspaces in $X$ are $C$-complementable, i.e. there exist projection with norm $\leq C$?

3) which subspaces in $X$ are $C$-complementable for all $C>1$?

4) I think that spaces $\ell_{1,0}(S')$ with $S'\subset S$ will fit. But are there other examples?

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1 Answer 1

up vote 4 down vote accepted

Just some simple remarks:

  1. If $P$ is projection on $X$, then $P$ extends to a projection on the completion $\ell_(S)$ with the same norm. Therefore, if $P$ has norm one, the closure of its range in $\ell_(S)$ is the closed span of disjointly supported functions. The only way this can happens is for the range of $P$ itself to be linearly spanned by disjointly supported functions and hence be isometrically isomorphic to $\ell_1(T)$ for some set $T$.

  2. From the first sentence of (1.) you get than any complemented subspace of $X$ has its completion isomorphic to $\ell_1(T)$ for some set $T$.

  3. Any subspace of $X$ spanned by disjointly supported functions is norm one complemented. That answers 4).

  4. A subspace of $X$ is $C$-complementable for all $C>1$ iff it is $1$-complementable iff it is spanned by disjointly supported functions. Again, just pass to the closure in $\ell_1(S)$ and use the known theory there, which I presume you know something about, else why this question?

Finally, why DID you ask this question?

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2  
+1, mainly for the final sentence. –  Yemon Choi Apr 2 '12 at 20:03
    
For what it's worth: my guess, which the OP can hopefully confirm or refute, is that this arises from work of Helemskii looking at projective objects in the category of normed spaces and contractive linear maps, relative to the class of all surjective contractive linear maps. It is not clear to me what the OP is doing here, though (study? writing a paper? etc.) –  Yemon Choi Apr 2 '12 at 20:05
    
Dear Bill Johnson thanks for your answer! I'm asking this question after reading about Grothendieck theorem describing projective Banach spaces. I'm a layman in advanced theory of Banach spaces and don't understand why 1) image of $\overline{Im(P)}$ must be closure of linear span of disjointly supported functions. As far as I understand $T$ has a cardinality of disjoint supports. 2) why $C$-complmentability for all $C>1$ implies 1-complementability. So could you recommend books necessary to understand, or explain it in your post? –  no identity Apr 2 '12 at 20:23
    
@Yemon Choi, yes my question also rised from work of Helemskii. The question is wheter extremal projectivity coincides with metric (if you familiar with this notions) –  no identity Apr 2 '12 at 20:26
2  
Look at Elton Lacey's book or volume two of Lindenstrauss-Tzafriri. Your setting is simpler (discrete rather than continuous). Contractively complemented subspaces of $L_1$ spaces must be sublattices. Almost contractively complemented subspaces of $L_1$, as in 4), are somewhat nastier. I don't remember and can't check now where they are treated in books. Your case is a bit simpler than the general, but I don't see an argument that is easy from "first principles". Maybe there is a simple direct argument that they must be contractively complemented and I just don't remember... –  Bill Johnson Apr 2 '12 at 20:53

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