Double-dual N-types over Banach Spaces Not Containing 1

Let E be a Banach space. The concept of n-type over E is introduced here, generalizing the concept of type over E introduced by Krivine and Maurey. Let E be the second dual of E and fix g i for all x ∈ E and all a 1 ,...,a n ∈ R, defines an n-type over E. Types that can be represented in this way are called double-dual n-types; we say that (g 1 ,...,g n) ∈ (E) n realizes τ. Let E be a (not necessarily separable) Banach space that does not contain 1. We study the set of elements of (E) n that realize a given double-dual n-type over E. We show that the set of realizations of this n-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 space E is 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. 1. Introduction. We first give a definition of n-types over Banach spaces and show how this definition generalizes the definition of type given by Krivine and Maurey.


Introduction.
We first give a definition of n-types over Banach spaces and show how this definition generalizes the definition of type given by Krivine and Maurey.
For every x = (x 1 ,...,x n ) ∈ E n , define τ x : E × R n → R by setting τ x y,a 1 ,...,a n = y + n i=1 a i x i (1.1) for all y ∈ E and for all a 1 ,...,a n ∈ R.
Definition 1.1.Let E be a Banach space and fix n ∈ N.For every x ∈ E n , let τ x be defined as above.A function τ : E → R is an n-type over E if τ is a function in the closure (with respect to the topology of pointwise convergence) of the set of functions {τ x : x ∈ E n }. [3] defined types over a Banach space E in the following way.For every x ∈ E, let t x : E → R be defined by t x (y) = y + x for all y ∈ E. Then a type over E is a function t : E → R in the closure (with respect to the topology of pointwise convergence) of the set {t x : x ∈ E}.

Krivine and Maurey
The types introduced by Krivine and Maurey coincide with the 1-types introduced in Definition 1.1 above.Indeed, every 1-type over E, τ defines a type (in the sense of Krivine and Maurey) by letting t(x) = τ(x, 1) for all x ∈ E. Conversely, if t is a type (in the sense of Krivine and Maurey), define τ(x, a) = |a|t((1/a)x) if a ≠ 0 and τ(x, 0) = x .With this definition, τ is a 1-type over E.
The definition of n-type over a Banach space E given above reflects an analyst's view of an n-type as a description of an n-tuple of elements (u 1 ,...,u n ) from a Banach space ultrapower of E. This notion of n-type coincides with the model theorist's notion of quantifier-free n-type over E in the language of Banach spaces.The reader is referred to [2] for more details.
Let E denote the second dual of E and let ḡ = (g 1 ,...,g n ) ∈ (E ) n .Define τ ḡ : E × R → R by setting τ ḡ (y, a 1 ,...,a n ) = y + n i=1 a i g i for all y ∈ E and all a 1 ,...,a n ∈ R. By the principle of local reflexivity, the function τ ḡ is an n-type over E. Types that can be realized in this way are called double-dual n-types over E.
Suppose that A ⊆ E and ḡ = (g 1 ,...,g n ) ∈ (E ) n .We let tp( ḡ /A) denote the function τ : Let τ be a double-dual n-type over E. Following the notation introduced in [1], we let be the set of elements of (E ) n that realize τ.

Statement of the main theorem.
The purpose of this paper is to prove the following theorem.
is convex.

3.
Lemmas.The proof of Proposition 2.2 requires 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 that E is a Banach space that does not contain 1 .We denote by E its dual and by E its second dual.If F is any Banach space and G ⊆ F , we denote by σ (F,G) the topology on F induced by open sets of the form {x ∈ F | x, g i ≤ ε for all i = 1,...,n}, where n ∈ N, g 1 ,...,g n ∈ G, and ε > 0. The closure of a set V ⊆ F with respect to this topology is denoted by σ (F,G)cl(V ).The weak * -topology on E is therefore denoted by σ (E ,E ).For brevity, we write If F is a normed space and M > 0, we let Let τ be a double-dual 1-type over E. Let S = Rep[τ] and g 1 ,g 2 ∈ S. In order to prove that S is convex, we need to show that the line segment joining g 1 and g 2 is contained in S. Set M = g 1 .
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 the ᏸ-structure (E, E ,E ) whose sorts are R, E, E , and E and whose functions are addition, scalar multiplication and the norm for each sort, the absolute value function for real numbers, the constants g 1 and g 2 , and the following additional functions: (3.1) Lemma 3.1.Let A c be a separable subset of E. There exists a separable approximate elementary substructure of ( E, E ,E ), (A,B,C) A (E, E ,E ), such that A c ⊆ A and for all c ∈ C and all δ > 0, the following condition holds: and d(c 0 ) = 0. Furthermore, for any approximate elementary substructure of (E, E ,E ), the following holds: (ii) if d(c) > 0, then there exists a ∈ A with a + g 1 ≠ a + c .
Proof.Using the downward Löwenheim-Skolem theorem [2, Theorem 9.14], choose a separable approximate elementary substructure of (E, E ,E ), There exist e 1 ,e 2 ,... ∈ E such that c j − e j ≤ 2d c j , d e j = 0, ( for all j ∈ N. There exists another separable approximate elementary substructure of (E, E ,E ), which contains (A 0 ,B 0 ,C 0 ∪{e 1 ,e 2 ,...}).We continue in this fashion through countably many steps and then take 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 ).Suppose (A,B,C) is an approximate elementary substructure of ( E, E ,E Here, the variable x ranges over the sort associated with E. Condition (ii) follows because the same formula is approximately true in (A,B,C).   (3.

Lemma 3.2. Let (A,B,C) A (E, E ,E ) as in
(3.9) Thus, P ≥ 1.For each b ∈ B, P b is the restriction of b to A and we obtain P b ≤ 1.
The following lemma is not needed but is of its own interest.

Lemma 3.3. Let (A,B,C) A (E, E ,E ) as in Lemma 3.1. Let U denote the unit ball of A and let V denote the unit ball of P B. Then V is σ (A ,A)-dense in U.
Proof.Let V = σ (A ,A)cl(V ).Lemma 3.2 and the weak * -lower semicontinuity of the norm yield V ⊆ U .Suppose V ≠ U. Then there exists b 0 ∈ U \ V .Since V is convex and weak * -closed (i.e., σ (A ,A)-closed) and {b 0 } is σ (A ,A)-compact, there exist a weak *continuous linear functional a of norm at most 1 and real numbers r < s such that for for all b ∈ V .Because a is a weak * -continuous linear functional on A , we see that a ∈ Ꮾ 1 (A).But then where r < s ≤ 1 are as before.Here, the variable a ranges over the sort associated with A and b ranges over the sort associated with B. We may then choose ε > 0 such that we obtain Fix such an element a ∈ E. We obtain This is a contradiction.
Let (x k ) k∈N enumerate a dense set in A and {λ 1 ,λ 2 ,...} enumerate a dense set in R. Without loss of generality, x 0 = 0, λ 0 = 0, and λ 1 = 1.(A,B,C) A (E, E ,E ) as in Lemma 3.1 and let P be as given by Lemma 3.2.For i = 1, 2, there exists a bounded sequence (a i,j ) j∈N in A such that lim j→∞ a i,j − g i ,b = 0 for all b ∈ B and

Lemma 3.4. Let
(3.17) A consequence of the Hahn-Banach theorem (see [5, Section 15, Lemma II.E., pages 76-79]) yields that k 1 ,k 2 ,k 3 ≤j U k 1 ,k 2 ,k 3 ,j is not empty and Therefore, there exists (e 1,j ,e 2,j ) Let y 1 ,y 2 be variables that range over the sort associated with E. For all k 1 ,k 2 ,k 3 ≤ j, let φ k 1 ,k 2 ,k 3 ,j (y 1 ,y 2 ) be the positive bounded ᏸ(A,B,C)-formula For all k ≤ j, let ψ k,j (y 1 ,y 2 ) be the positive bounded ᏸ(A,B,C)-formula Recall that M = g 1 .By the initial observation, we have Because (A,B,C) A (E, E ,E ), we have We may therefore choose a 1,j and a 2,j in A such that , 2 and all k ≤ j.Further, a i,j ≤ M + 1.Thus, the sequences (a i,j ) j∈N are bounded for i = 1, 2. The statement of the lemma follows because {b 1 ,b 2 ,...} is dense in B and (a i,j ) j∈N is bounded for i = 1, 2.
The hypothesis that E does not contain 1 has not been used so far.In the following two lemmas, we make use of this hypothesis.

Lemma 3.5. Let (A,B,C) A (E, E ,E ) as in Lemma 3.1. There exists an isometric embedding
Proof.Using Lemma 3.4, there exists, for each i = 1, 2, a bounded sequence (a i,j ) j∈N in A such that lim j→∞ a i,j − g i ,b = 0 for all b ∈ B and 1 and the sequence (a i,j ) j∈N is bounded, we may apply Rosenthal's theorem [4].We obtain a function j : N → N with m ≤ j(m) < j(m + 1) for all m ∈ N such that the subsequence (a i,j(m) ) m∈N is σ (A,A )-Cauchy for each i = 1, 2. We then define, for every i = 1, 2, a linear functional Ψ i ∈ A by setting Ψ i (a ) = lim m→∞ a i,j(m) ,a for all a ∈ A .We then define a linear operator Q : span(A ∪ {g 1 ,g 2 }) → A by setting Qg i = Ψ i for i = 1, 2 and Qa = Ja for all a ∈ A. (Here, J : A → A denotes the canonical embedding from A into A .) It is immediate from the definition that Q fixes IA pointwise and P b, Q(Ia + λg 1 + µg 2 ) = b, Ia + λg 1 + µg 2 for all a ∈ A, b ∈ B, and λ, µ ∈ R.
We show that Q is an isometry.Let λ, µ ∈ R and a ∈ A. Because P (B) ⊆ A , we have (3.25) On the other hand, the norm is σ (A ,A )-lower semicontinuous.Thus, for all integers k 1 ,k 2 ,k 3 ∈ N, we have . We show that g ∈ S: suppose that g ∈ S. Then d(g ) > 0, and by Lemma 3.1(ii), there exists a ∈ A such that a + g ≠ a + g 1 .But then a + c = a + Qg = a + g ≠ a + g 1 , which contradicts the assumption that c ∈ Rep[tp(g 1 /A)].Therefore, g ∈ S ∩ C 0 , and so We obtain The following lemma shows that S contains all convex combinations of g 1 and g 2 .Because g 1 and g 2 are arbitrary elements of S = Rep[τ], this shows that Rep[τ] is convex.
Lemma 3.6.Let (A,B,C) A (E, E ,E ) as in Lemma 3.1 and let Q be as given by Lemma 3.5.Then Q(C ∩ S) contains all convex combinations of Qg 1 and Qg 2 , and S contains all convex combinations of g 1 and g 2 .
Proof.Let λ ∈ [0, 1].By construction, the signature has constants g 1 and g 2 of the sort associated with E .Therefore, g 1 ,g 2 ∈ C. Since g 1 ,g 2 ∈ S, we obtain g 1 ,g 2 ∈ S ∩C.By the previous remark, Q(g 1 ) and Q(g 2 ) are elements of Rep[tp(g 1 /A)].Since A is separable, Theorem 2.3 yields that Rep[tp(g 1 /A)] is convex.Therefore We are now ready to prove Proposition 2.2.
Proof.Let E be a Banach space that does not contain 1 , and let τ be a double-dual 1-type over E. Let g 1 ,g 2 ∈ S = Rep[τ].Let A 0 ⊆ E be any separable set.Then choose an approximate elementary substructure (A,B,C) (E, E ,E ) as in Lemma 3.1.Choose the isometric embedding Q as in Lemma 3.5.By Lemma 3.6, S = Rep[τ] contains all linear combinations of the form λg 1 +(1−λ)g 2 .Because g 1 ,g 2 ∈ Rep[τ] were arbitrary, we obtain that Rep[τ] is convex.

Remarks and questions.
We conclude this paper by remarking that the hypothesis that E does not contain 1 in Proposition 2.2 cannot be removed.Indeed, let E = 1 .Choose g ∈ E in the band ⊥⊥ 1 with g = 1.Then τ g (x) = x +g = x + g = x + 1 and τ −g (x) = x − g = x + g = x + 1 for all x ∈ E. Therefore, −g ∈ Rep[τ g ], which shows that Rep[τ g ] is not convex.

Lemma 3. 1 .
There exists an isometric embedding P : B → A such that a, P b = a, b for all a ∈ A and all b ∈ B. Proof.Define P : B → A by setting a, P b = a, b for all b ∈ B and all a ∈ A. The function P is linear; we need to show that it is an isometry.Let 1 > ε > 0. Observe that 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 variable b ranges over B and a ranges over A. Now, fix b ∈ B of norm 1 and set b 0 = (1 − ε)b.Consider the sentence

Theorem 2.3 (Haydon and Maurey). Let E be a separable Banach space that does not contain 1 . Let τ be a double-dual 1-type over E. Then Rep[τ] is convex. The proof provided in [1] requires the hypothesis that E is separable. The purpose of this paper is to show how methods from model theory can be used to remove
Theorem 2.1.Let E be a Banach space that does not contain 1 .Let τ be a doubledual n-type over E. Then Rep[τ] is convex.If we take n = 1, we obtain the following proposition.Proposition 2.2.Let E be a Banach space that does not contain 1 .Let τ be a doubledual 1-type over E. Then Rep[τ] is convex.