A proof of the seminal result of B. Maurey, that reflexive subspaces of L_{1}[0, 1] enjoy the fpp .

Proposition. If X is a reflexive subspace of L_{1}[0, 1] then X has the fpp .

Proof. First note that, since X is a reflexive, hence superreflexive [??], subspace of L_{1}[0, 1], its ultrapower (X)_{U} is a reflexive subspace of (L_{1}[0, 1])_{U} and (L_{1}[0, 1])_{U} is itself of the form L_{1}(Ω, Σ, μ), for some measure space (Ω, Σ, μ), [A-K] or [S82].

SHOULD HAVE INCLUDED A PROOF OF THIS LAST STATEMENT SOME-WHERE? (IT WOULD BE A BIT OF A DETOUR, BUT MAKES THINGS MORE SELF CONTAINED)

Now, suppose that X fails to have the fpp . Then, as described prior to proposition 2, in (X)_{U} we can construct a set C̃, which is weak compact (proposition 2), and a nonexpansive mapping T̃ of C̃ into itself with the properties discussed above. We will show that this implies X contains isometric copies of ℓ_{1}^{n} for arbitrarily large finite dimensions n, and hence (X)_{U} contains an isometric copy of ℓ_{1} and so can not be reflexive, a contradiction which will establish the result.

Using lemma 4 and remark 5 we can find fixed points f_{0}, f_{1}, ⋯, f_{2m} of T̃ in C̃ such that

∥f_{0} − f_{2m}∥_{1} = 1 and ∥f_{k} − f_{k−1}∥_{1} = 1/2^{m}

Point-wise we have almost everywhere,

|f_{0}(ω) − f_{2m}(ω)| ≤ ∑_{k=1}^{2m} |f_{k}(ω) − f_{k−1}(ω)|,

while the L_{1}-norm of the left hand side equals the L_{1}-norm of the right hand side so

(*)  |f_{0}(ω) − f_{2m}(ω)| = ∑_{k=1}^{2m} |f_{k}(ω) − f_{k−1}(ω)|, a.e.

Claim 1: For each k the set {ω : f_{k}(ω) ≠ f_{0}, or f_{k}(ω) ≠ f_{2m}} is a null set.

Consequently we have, |f_{k} − f_{k−1}| ≡ |f_{0} − f_{2m}|χ_{A_{k}}, for some measurable set A_{k}. By (*) the A_{k} may be taken to be mutually disjoint. Thus,

(2^{m}[f_{k} − f_{k−1}])_{k=1}^{2m}

are normalized disjointly supported elements of (X)_{U} and hence isometrically span ℓ_{1}^{2m}, providing the sought after contradiction from which the result follows.

Proof of claim 1: Suppose this were not the case, then there exists a k such that {ω : f_{k}(ω) ≠ f_{0}, or f_{k}(ω) ≠ f_{2m}} has non zero measure. But then, at least one of the following six possibilities must hold on a set B of strictly positive measure:

Typeset by AMS-TEX

(1) f_{0} < f_{k} < f_{2m},

(2) f_{0} < f_{2m} < f_{k},

(3) f_{k} < f_{0} < f_{2m},

(4) f_{k} < f_{2m} < f_{0},

(5) f_{2m} < f_{0} < f_{k},

(6) f_{2m} < f_{k} < f_{0}. Let us assume it is case(1), the argument for the other cases is similar. Claim 2: There exist x, y ∈ C with J x(ω) ≤ f_{k}(ω) ≤ J y(ω) for almost all ω in B. But then,

|f_{k}(ω) − J (\frac{x + y}{2} ) (ω)| ≤ \frac{1}{2} |f_{k}(ω) − J x(ω)| + \frac{1}{2} |f_{k}(ω) − J y(ω)|

= \frac{1}{2} (f_{k}(ω) − J x(ω)) + \frac{1}{2} (J y(ω) − f_{k}(ω))

< J y − J x, for almost all ω ∈ B.

Whence, by the Generalized Goebel-Karlovitz lemma,

1 = diam C = ∥f_{k} − J (\frac{x + y}{2} )∥_{1} < ∥x − y∥_{1} ≤ 1,

a contradiction which establishes claim 1. It therefore only remains to prove claim 2. Proof of claim 2:

THIS IS WHERE I RAN OUT OF STEAM. NEEDS THE RELEVANT PIECE OF ARGUMENT FROM [E] INSERTED

□ Using similar arguments Maurey [M] also established the w–fpp for the Hardy

space H_{1}. [[M]] Maurey, B., Points fixes des contractions de certains faiblement compacts de L^{1},

Seminaire d’Analyse Fonctionnelle, Exposé No. VIII (1980), 18.