A NULLSTELLENSATZ FOR AMOEBAS

514 JOHNSON and ZHENG
where C is the unconditional constant associated with the USB FDD (E n ).Ifwelet
∑
δi
1 − ɛ
C(1 + ɛ) .
Therefore (xi ∗ ) is unconditional with unconditional constant less than (1 + 3ɛ)C if ɛ
is sufficiently small. Hence (Fn ∗ ) is a USB FDD.
THEOREM 2.13
Let X be a separable reflexive Banach space. Then the following are equivalent.
(a) X has the UTP.
(b) X embeds into a reflexive Banach space with a USB FDD.
(c) X ∗ has the UTP.
Proof
It is obvious that (b) implies (a). If we can prove that (a) implies (b) and that X satisfies
(b), then by Lemma 2.12, X ∗ is a quotient of a reflexive space with a USB FDD. So,
by Theorem 2.8, X ∗ has the UTP. Hence we need only show that (a) implies (b). Let
X 1 be a subspace of X with an FDD (E n ) given by Lemma 2.11.By[13, Proposition
2.4], we get a blocking (F n ) of (E n ) so that (F n ) is a USB FDD. Let Y 1 = [F 4n ],and
let Y 2 = [F 4n+2 ].Then(F 4n ) and (F 4n+2 ) form UFDDs for Y 1 and Y 2 . By Lemma 2.11,
X/Y i has an FDD. Since X has the UTP, by Theorem 2.8 we know that X/Y i has
the UTP. By using [13, Proposition 2.4] again, we know that X/Y i has a USB FDD.
Noticing that Y 1 ∩ Y 2 ={0} and that Y 1 + Y 2 is closed, by Lemma 2.10 we have that
X embeds into X/Y 1 ⊕ X/Y 2 . Hence X embeds into a reflexive space with a USB
FDD.
COROLLARY 2.14
Let X be a separable reflexive Banach space with the UTP. Then X embeds into a
reflexive Banach space with an unconditional basis.
Proof
By Theorem 2.13, X embeds into a reflexive space Y with a USB FDD (E n ).We
prove that Y embeds into a reflexive space with a UFDD. Then, as mentioned in the
introduction, Y embeds into a reflexive space with an unconditional basis, and so X
does, too.
By Lemma 2.12, there is a blocking (F n ) of (E n ) such that (Fn ∗ ) is a USB FDD for
Y ∗ .Now,letY 1 = ⊕ F 4n ,andletY 2 = ⊕ F 4n+2 .ThenwehaveY 1 ∩ Y 2 ={0}, and
Y 1 + Y 2 is closed because (F 2n ), being a skipped blocking of (E n ), is unconditional.
By Lemma 2.10, Y embeds into Y/Y 1 ⊕ Y/Y 2 .Since(Y/Y i ) ∗ is isomorphic to Yi
⊥ ,it