PII. S0161171204211152 http://ijmms.hindawi.com © Hindawi Publishing Corp.
DOUBLE-DUAL
n
-TYPES OVER BANACH SPACES
NOT CONTAINING
1
MARKUS POMPER
Received 6 November 2002
LetEbe a Banach space. The concept ofn-type overEis introduced here, generalizing the concept oftype over Eintroduced by Krivine and Maurey. LetEbe the second dual ofE
and fixg1, . . . , gn∈E. The functionτ:E×Rn→R, defined by lettingτ(x, a1, . . . , an)= x+n
i=1aigifor allx∈Eand alla1, . . . , an∈R, defines ann-type overE. Types that can be represented in this way are called double-dualn-types; we say that(g1, . . . , gn)∈(E)n realizesτ. LetEbe a (not necessarily separable) Banach space that does not contain1. We study the set of elements of(E)nthat realize a given double-dualn-type overE. We show that the set of realizations of thisn-type is convex. This generalizes a result of Haydon and Maurey who showed that the set of realizations of a given 1-type over a separable Banach spaceEis convex. The proof makes use of Henson’s language for normed space structures and uses ideas from mathematical logic, most notably the Löwenheim-Skolem theorem.
2000 Mathematics Subject Classification: 46B04, 46B20.
1. Introduction. We first give a definition ofn-types over Banach spaces and show how this definition generalizes the definition oftypegiven by Krivine and Maurey.
For every ¯x=(x1, . . . , xn)∈En, defineτ ¯
x:E×Rn→Rby setting
τ¯x
y, a1, . . . , an=y+ n
i=1
aixi (1.1)
for ally∈Eand for alla1, . . . , an∈R.
Definition1.1. LetE be a Banach space and fixn∈N. For every ¯x∈En, letτ¯ x
be defined as above. A functionτ:E→Ris ann-type over Eifτis a function in the closure (with respect to the topology of pointwise convergence) of the set of functions
{τ¯x: ¯x∈En}.
Krivine and Maurey [3] defined types over a Banach spaceEin the following way. For everyx∈E, let tx:E→Rbe defined bytx(y)= y+xfor ally∈E. Then a type overEis a functiont:E→Rin the closure (with respect to the topology of pointwise convergence) of the set{tx:x∈E}.
The types introduced by Krivine and Maurey coincide with the 1-types introduced in
The definition ofn-type over a Banach spaceEgiven above reflects an analyst’s view of an n-type as a description of ann-tuple of elements (u1, . . . , un) from a Banach space ultrapower ofE. This notion ofn-type coincides with the model theorist’s notion of quantifier-freen-type overEin the language of Banach spaces. The reader is referred to [2] for more details.
LetEdenote the second dual ofEand let ¯g=(g1, . . . , gn)∈(E)n. Defineτg¯:E× R→Rby settingτ¯g(y, a1, . . . , an)= y+ni=1aigifor ally∈Eand alla1, . . . , an∈R.
By the principle of local reflexivity, the functionτg¯is ann-type overE. Types that can
be realized in this way are calleddouble-dualn-types overE.
Suppose thatA⊆Eand ¯g=(g1, . . . , gn)∈(E)n. We let tp(g¯/A)denote the
func-tionτ:A×Rn→Rdefined by settingτ(x, a1, . . . , an)= x+n
i=1aigifor allx∈A
and alla1, . . . , an∈R.
Letτbe a double-dualn-type overE. Following the notation introduced in [1], we let
Rep[τ]=g¯∈(E)n:τ=τg¯
(1.2)
be the set of elements of(E)nthat realizeτ.
2. Statement of the main theorem. The purpose of this paper is to prove the fol-lowing theorem.
Theorem2.1. LetEbe a Banach space that does not contain1. Letτ be a double-dualn-type overE. ThenRep[τ]is convex.
If we taken=1, we obtain the following proposition.
Proposition2.2. LetEbe a Banach space that does not contain1. Letτbe a double-dual1-type overE. ThenRep[τ]is convex.
The previous proposition is a generalization of a result of Haydon and Maurey [1, Theorem 3.2].
Theorem2.3(Haydon and Maurey). LetEbe a separable Banach space that does not contain1. Letτbe a double-dual1-type overE. ThenRep[τ]is convex.
The proof provided in [1] requires the hypothesis thatEis separable. The purpose of this paper is to show how methods from model theory can be used to remove this hypothesis.
The proof ofProposition 2.2will be provided inSection 3. The following proposition shows thatTheorem 2.1is an immediate consequence ofProposition 2.2.
Proposition2.4. LetEbe a Banach space. Suppose thatRep[τ]is convex whenever
τ is a double-dual 1-type overE. ThenRep[τ]is convex wheneverτ is a double-dual
n-type overE.
Proof. Letτbe a double-dualn-type overEand suppose thatτis realized in(E)n
by(g1, . . . , gn). For any givenn-tuple(a1, . . . , an)∈Rn, consider the function
D(a1,...,an):(E)n →E,
u1, . . . , un
→ n
i=1
aiui. (2.1)
Therefore,D−1
(a1,...,an)(Rep[tp( n
i=1aigi/E)])is convex for alla1, . . . , an∈R. Thus
Reptpg1, . . . , gn =
(a1,...,an)∈Rn
D−(a11,...,an) Rep
tp
n
i=1 aigi/E
(2.2)
is convex.
3. Lemmas. The proof ofProposition 2.2requires a sequence of lemmas, which will be discussed in this section. The proof of the proposition will be provided at the end of this section.
Throughout this section, we assume thatEis a Banach space that does not contain1. We denote byEits dual and byEits second dual. IfFis any Banach space andG⊆F, we denote byσ (F , G) the topology onF induced by open sets of the form{x∈F | x, gi ≤εfor alli=1, . . . , n}, wheren∈N,g1, . . . , gn∈G, andε >0. The closure of a setV⊆Fwith respect to this topology is denoted byσ (F , G)cl(V ). The weak∗-topology onE is therefore denoted byσ (E, E). For brevity, we writeV=σ (E, E)cl(V )for anyV⊆E.
If F is a normed space andM >0, we letᏮM(F )= {x∈F | x ≤M}denote the closedM-ball inF.
Letτbe a double-dual 1-type overE. LetS=Rep[τ]andg1, g2 ∈S. In order to prove
thatSis convex, we need to show that the line segment joiningg1andg2 is contained inS. SetM= g1.
We will use Henson’s language for normed space structures. See [2] for more details. It is assumed that the reader is familiar with the concepts of normed space structures [2, Sections 2 and 3], positive bounded formulas (Section 5), approximate satisfaction and approximate elementary substructures (Section 6), and the Löwenheim-Skolem theorem (Section 9).
Consider theᏸ-structure(E, E, E)whose sorts areR,E,E, andEand whose func-tions are addition, scalar multiplication and the norm for each sort, the absolute value function for real numbers, the constantsg1 andg2, and the following additional func-tions:
E →E, x→I(x):=the canonical image ofxinE,
E →R, x→d(x):=infx−s:s∈S,
E×E →R, (x, x)→ x, x,
E×E →R, (x, x)→ x, x.
(3.1)
Lemma3.1. LetAc be a separable subset ofE. There exists a separable approximate elementary substructure of(E, E, E),(A, B, C)A(E, E, E), such thatAc⊆Aand for allc∈Cand allδ >0, the following condition holds:
(i) (A, B, C)Ad(c)≤δimplies that there exists ac0∈C such thatc0−c ≤2δ
andd(c0)=0.
Furthermore, for any approximate elementary substructure of(E, E, E), the follow-ing holds:
Proof. Using the downward Löwenheim-Skolem theorem [2, Theorem 9.14], choose a separable approximate elementary substructure of(E, E, E),
A0, B0, C0AE, E, E, (3.2)
such thatAc⊆A. Letc1, c2, . . .enumerate a dense subset ofC0. There existe1, e2, . . .∈E
such that
cj−e j≤2d
cj, dej
=0, (3.3)
for allj∈N.
There exists another separable approximate elementary substructure of(E, E, E),
A1, B1, C1AE, E, E, (3.4)
which contains(A0, B0, C0∪{e1, e2, . . .}). We continue in this fashion through countably many steps and then take
(A, B, C)=cl
∞
k=1
Ak, Bk, Ck
. (3.5)
This structure is an approximate elementary substructure of (E, E, E) in which (i) holds.
Condition (ii) holds in every approximate elementary substructure of(E, E, E). Sup-pose(A, B, C)is an approximate elementary substructure of(E, E, E)andc∈Csuch thatd(c)=δ1>0. Thenc∈S=Rep[tp(g1/E)]. Therefore, there existe∈Eandδ2>0
such that|e+c−e+g1| ≥δ2. Letm= e+1. Hence
(E, E, E)A∃mxx+c−x+g1≤δ2
. (3.6)
Here, the variablex ranges over the sort associated withE. Condition (ii) follows be-cause the same formula is approximately true in(A, B, C).
Lemma3.2. Let(A, B, C)A (E, E, E)as in Lemma 3.1. There exists an isometric embeddingP:B→Asuch thata, P b = a, bfor alla∈Aand allb∈B.
Proof. DefineP:B→Aby settinga, P b = a, bfor allb∈Band alla∈A. The functionPis linear; we need to show that it is an isometry. Let 1> ε >0. Observe that
(A, B, C)A∀1bb ≤(1−ε)3∨∃1aa, b ≥(1−ε)4 (3.7)
because the same sentence is approximately true in the structure (E, E, E) by the definition of the norm of a linear functional. Here, the variablebranges overBanda
ranges overA. Now, fixb∈Bof norm 1 and setb0=(1−ε)b. Consider the sentence
Because this sentence is true in(A, B, C), there existsa∈Asuch thata ≤(1−ε)−1
anda, b0 ≥(1−ε)2. Then
P b =(1−ε)−1P b0≥
a, P b0 (1−ε)a≥
(1−ε)2
(1−ε)(1−ε)−1=(1−ε)
2. (3.9)
Thus,P ≥1. For eachb∈B,P bis the restriction ofbtoAand we obtainP b ≤1.
The following lemma is not needed but is of its own interest.
Lemma3.3. Let(A, B, C)A(E, E, E)as inLemma 3.1. LetUdenote the unit ball of
Aand letV denote the unit ball ofP B. ThenV isσ (A, A)-dense inU.
Proof. LetV=σ (A, A)cl(V ). Lemma 3.2and the weak∗-lower semicontinuity of the norm yieldV⊆U. SupposeV≠U. Then there existsb0∈U\V. SinceVis convex and weak∗-closed (i.e.,σ (A, A)-closed) and{b0}isσ (A, A)-compact, there exist a weak∗ -continuous linear functionalaof norm at most 1 and real numbersr < ssuch that for allb∈V,
a, b ≤r < s <a, b0≤1. (3.10)
SinceV is symmetric about the origin, we get
a, b≤r < s <a, b0≤1 (3.11)
for all b∈V. Becausea is a weak∗-continuous linear functional on A, we see that
a∈Ꮾ1(A). But then
(A, B, C)A∃1aa ≥s∧∀1b−r≤ a, b ≤r, (3.12)
wherer < s≤1 are as before. Here, the variablearanges over the sort associated with
Aandbranges over the sort associated withB. We may then chooseε >0 such that
(1+ε)3r < s. Setλ=(1+ε). Because
(E, E, E)A∃1aa ≥s∧∀1b−r≤ a, b ≤r, (3.13)
we obtain
(E, E, E) ∃λaa ≥λ−1s∧∀1/λb−λr≤ a, b ≤λr. (3.14)
Fix such an elementa∈E. We obtain
a =supa, b:b∈Ꮾ1(E)
=sup(1+ε)a, b:b∈Ꮾ(1+ε)−1(E)
≤(1+ε)2r < (1+ε)−1s≤ a.
(3.15)
Let(xk)k∈Nenumerate a dense set inAand{λ1, λ2, . . .}enumerate a dense set inR. Without loss of generality,x0=0,λ0=0, andλ1=1.
Lemma 3.4. Let (A, B, C)A (E, E, E) as in Lemma 3.1 and letP be as given by
Lemma 3.2. For i= 1,2, there exists a bounded sequence (ai,j)j∈N in A such that limj→∞ai,j−gi, b =0for all b∈B andxk1+λk2a1,j+λk3a2,j ≤ xk1+λk2g1+ λk3g2+2−j+1for all1≤k1, k2, k3≤j∈N.
Proof. Letb1, b2, . . .enumerate a dense set inB. Fixj ∈N. Letk1, k2, k3≤j and consider the set
Uk1,k2,k3,j=
e1, e2∈E2:xk
1+λk2e1+λk3e2<xk1+λk2g1+λk3g2+2−j
.
(3.16)
Observe thatUk1,k2,k3,jis an open convex set. Furthermore,(g1, g2)∈◦(Uk 1,k2,k3,j)for allk1, k2, k3≤j. Thus
g1, g2∈ k1,k2,k3≤j
◦ Uk1,k2,k3,j
. (3.17)
A consequence of the Hahn-Banach theorem (see [5, Section 15, Lemma II.E., pages 76– 79]) yields thatk1,k2,k3≤jUk1,k2,k3,j is not empty and
g1, g2
∈
◦∼
k1,k2,k3≤j
Uk1,k2,k3,j
. (3.18)
Therefore, there exists(e1,j, e2,j)∈k1,k2,k3≤jUk1,k2,k3,j such that|ei,j−gi, bk| ≤2−j
fori=1,2 and allk≤j.
Lety1, y2be variables that range over the sort associated withE. For allk1, k2, k3≤j, letφk1,k2,k3,j(y1, y2)be the positive boundedᏸ(A, B, C)-formula
xk1+λk2y1+λk3y2−xk1+λk2g1+λk3g2≤2−j. (3.19)
For allk≤j, letψk,j(y1, y2)be the positive boundedᏸ(A, B, C)-formula
y1−g1, bk≤2−j∧y2−g
2, bk≤2−j. (3.20)
Letρj(y1, y2)be the positive boundedᏸ(A, B, C)-formula
k1,k2,k3≤j
φk1,k2,k3,j
y1, y2∧ k≤j
ψk,jy1, y2. (3.21)
Recall thatM= g1. By the initial observation, we have
(E, E, E)A∃M+1y1∃M+1y2ρjy1, y2. (3.22)
Because(A, B, C)A(E, E, E), we have
We may therefore choosea1,j anda2,jinAsuch that
xk1+λk2a1,j+λk3a2,j<xk1+λk2g1+λk3g2+2−j+1 (3.24)
for all k1, k2, k3≤j and |ai,j−gi, bk| ≤2−j+1 for i=1,2 and all k≤j. Further, ai,j ≤M+1. Thus, the sequences(ai,j)j∈N are bounded fori=1,2. The statement of the lemma follows because {b1, b2, . . .}is dense inB and(ai,j)j∈N is bounded for
i=1,2.
The hypothesis thatEdoes not contain1has not been used so far. In the following two lemmas, we make use of this hypothesis.
Lemma3.5. Let(A, B, C)A (E, E, E)as in Lemma 3.1. There exists an isometric embeddingQ: span(A∪ {g1, g2})→A such thatP b, Q(Ia+λg1+µg2) = b, Ia+ λg1+µg2for alla∈A,b∈B, andλ, µ∈R.
Proof. UsingLemma 3.4, there exists, for eachi=1,2, a bounded sequence(ai,j)j∈N
in Asuch that limj→∞ai,j−gi, b =0 for allb∈B and xk1+λk2a1,j+λk3a2,j ≤
xk1+λk2g1+λk3g2+2−j+1for all 1≤k1, k2, k3≤j∈N. BecauseAdoes not contain 1and the sequence(ai,j)j∈Nis bounded, we may apply Rosenthal’s theorem [4]. We obtain a functionj:N→Nwith m≤j(m) < j(m+1)for all m∈Nsuch that the subsequence(ai,j(m))m∈Nisσ (A, A)-Cauchy for eachi=1,2. We then define, for every
i=1,2, a linear functionalΨi∈Aby settingΨi(a)=limm→∞ai,j(m), afor alla∈A. We then define a linear operatorQ: span(A∪ {g1, g2})→A by settingQgi=Ψifor i=1,2 andQa=Jafor alla∈A. (Here,J:A→A denotes the canonical embedding fromAintoA.)
It is immediate from the definition thatQfixesIApointwise andP b, Q(Ia+λg1+ µg2) = b, Ia+λg1+µg2for alla∈A,b∈B, andλ, µ∈R.
We show thatQis an isometry. Letλ, µ∈Randa∈A. BecauseP (B)⊆A, we have
Qλg1+µg2+a=sup
Qλg1+µg2+a
, a:a∈Ꮾ1(A) ≥supQλg1+µg2+a
, a:a∈Ꮾ1P (B) =supλg1+µg2+a, b:b∈Ꮾ1(B)
=λg1+µg2+a.
(3.25)
On the other hand, the norm isσ (A, A)-lower semicontinuous. Thus, for all integers
k1, k2, k3∈N, we have
Q
xk1+λk2g1+λk3g2≤lim infm→∞ Q
xk1+λk2a1,j(m)+λk3a2,j(m)
≤lim inf
m→∞ xk1+λk2a1,j(m)+λk3a2,j(m)+2−
j(m)+1
≤xk1+λk2g1+λk3g2.
(3.26)
Therefore,Qis an isometry.
someg∈S∩C0. Then, for everya∈A, we have
a+c = a+g =a+g1. (3.27)
So,Q(S∩C0)⊆Rep[tp(g1/A)]∩Q(C0).
Conversely, Q(S∩C0)⊇Rep[tp(g1/A)]∩Q(C0). Indeed, if c∈ Rep[tp(g1/A)]∩ Q(C0), there existsg∈C0such thatQ(g)=c. We show thatg∈S: suppose that
g∈S. Thend(g) >0, and byLemma 3.1(ii), there existsa∈Asuch thata+g≠ a+g1. But then a+c = a+Qg = a+g≠a+g1, which contradicts
the assumption thatc∈Rep[tp(g1/A)]. Therefore,g∈S∩C0, and soc=Q(g)∈ Q(S∩C0).
We obtain
QS∩C0=Reptpg1/A ∩Q
C0. (3.28)
The following lemma shows thatScontains all convex combinations ofg1andg2. Becauseg1 andg2 are arbitrary elements ofS =Rep[τ], this shows that Rep[τ] is
convex.
Lemma 3.6. Let(A, B, C)A(E, E, E) as in Lemma 3.1and let Qbe as given by
Lemma 3.5. ThenQ(C∩S)contains all convex combinations ofQg1 andQg2, andS
contains all convex combinations ofg1andg2.
Proof. Letλ∈[0,1]. By construction, the signature has constantsg1andg2 of the
sort associated withE. Therefore,g1, g2∈C. Sinceg1, g2 ∈S, we obtaing1, g2∈ S∩C. By the previous remark,Q(g1)andQ(g2)are elements of Rep[tp(g1/A)]. Since
Ais separable,Theorem 2.3yields that Rep[tp(g1/A)]is convex. Therefore
Qλg1+(1−λ)g2
=λQg1+(1−λ)Qg2∈Rep
tpg1/A ∩Q
C0. (3.29)
By (3.28),
λg1+(1−λ)g2∈S∩C0⊆Rep
tpg1/E . (3.30)
We are now ready to proveProposition 2.2.
Proof. LetEbe a Banach space that does not contain1, and letτbe a double-dual 1-type overE. Letg1, g2∈S=Rep[τ]. LetA0⊆Ebe any separable set. Then choose an approximate elementary substructure(A, B, C)(E, E, E)as inLemma 3.1. Choose the isometric embeddingQas inLemma 3.5. ByLemma 3.6,S=Rep[τ]contains all lin-ear combinations of the formλg1+(1−λ)g2. Becauseg1, g2∈Rep[τ]were arbitrary, we obtain that Rep[τ]is convex.
Haydon and Maurey also proved that Rep[τ]is compact with respect to the weak∗ -topology ifEis separable andτis a double-dual 1-type overE.
This poses the following question.
Question4.1. LetE be a (not necessarily separable) Banach space that does not contain1. Letτbe a double-dual 1-type overE. Is then Rep[τ]compact with respect to the weak∗-topology onE?
References
[1] R. Haydon and B. Maurey,On Banach spaces with strongly separable types, J. London Math. Soc. (2)33(1986), no. 3, 484–498.
[2] C. W. Henson and J. N. Iovino,Ultraproducts in analysis, Analysis and Logic (Mons, 1997), London Math. Soc. Lecture Note Ser., vol. 262, Cambridge University Press, Cambridge, 2002, pp. 1–110.
[3] J.-L. Krivine and B. Maurey,Espaces de Banach stables, Israel J. Math.39(1981), no. 4, 273– 295.
[4] H. P. Rosenthal,A characterization of Banach spaces containingl1, Proc. Natl. Acad. Sci. USA
71(1974), 2411–2413.
[5] P. Wojtaszczyk,Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991.
Markus Pomper: Division of Natural Science and Mathematics, Indiana University East, Richmond, IN 47358, USA