- Email: [email protected]
On L∞ -envelopes of Banach spaces
J. Math. Anal. Appl. 394 (2012) 152–158
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
On L∞ -envelopes of Banach spaces Jesús M.F. Castillo a,∗ , Jesús Suárez b a
Departamento de Matemáticas, Facultad de Ciencias, Univ. de Extremadura, Avda. de Elvas s/n, 06071 Badajoz, Spain
b
Escuela Politécnica, Universidad de Extremadura, Avda. de la Universidad s/n, 10071 Cáceres, Spain
article
info
Article history: Received 24 June 2011 Available online 28 April 2012 Submitted by R.M. Aron Keywords: Extension of operators L∞ -spaces
abstract Let A denote any of the following classes of L∞ -spaces: C (K )-spaces, Lindenstrauss spaces, λ-separably injective spaces, universally λ-separably injective spaces, λ-Lindenstrauss– Pełczyński spaces or L∞,λ -spaces. We show that every Banach space X can be isometrically embedded into a space A(X ) ∈ A so that every operator X → A with A ∈ A can be extended to an operator A(X ) → A with the same norm. © 2012 Elsevier Inc. All rights reserved.
1. Introduction and preliminaries As is well-known, every Banach space X can be naturally embedded into the space of continuous functions C (BX ∗ ) in such a way that the embedding δX : X −→ C (BX ∗ ) enjoys the following universal property: Every C (K ) − valued operator τ : X → C (K ) can be extended through δX to an operator T : C (BX ∗ ) → C (K ) so that ∥T ∥ = ∥τ ∥. See [1], and also [2] for a historical account and a categorical presentation of this result. Does there exist a similar universal embedding for other classes of Banach spaces? It is particularly interesting to consider the situation replacing C (K )-spaces by their ‘‘local’’ versions, the L∞ spaces. Recall that a Banach space X is said to be an L∞,λ -space, λ ≥ 1, if every finite dimensional subspace F of X is contained in another finite dimensional subspace G of X whose Banach–Mazur distance to the corresponding ℓdimG is at most λ. A space X is said to be an L∞ -space if it is an L∞,λ -space for some λ ≥ 1. The ∞ following subclasses of L∞ -spaces have appeared in the literature (the missing definitions can be found in the next paragraph): L∞,λ -spaces, λ-Lindenstrauss–Pełczyński spaces, Lindenstrauss spaces, C (K )-spaces, λ-separably injective spaces, universally λ-separably injective spaces and λ-injective spaces. A Banach space is said to be a Lindenstrauss space if it is an L∞,1+ε -space for all ε > 0. The space X is said to be injective if every operator τ : Y → X can be extended to an operator T : W → X for any bigger superspace W of Y . The space X is said to be separably injective (resp. universally separably injective) [3] when the condition above is satisfied for separable W (resp. separable) Y . It is not hard to see [3] that for separable injectivity it is enough to take W = ℓ1 ; and W = ℓ∞ for universal separable injectivity. The space X is said to be a Lindenstrauss–Pełczyński space [4,5] when the previous condition holds for W = c0 . When an extension T verifying ∥T ∥ ≤ λ∥τ ∥ exists then the classes will be referred to, respectively, as λ-injective, λ-separably injective, universally λ-separably injective, and λ-Lindenstrauss–Pełczyński spaces. Definition 1.1. Let A be a class of Banach spaces. An A-envelope of a Banach space X is a couple (A(X ), δ) formed by a Banach space A(X ) ∈ A and an isometric embedding δ : X → A(X ) with the property that for every A ∈ A every operator τ : X → A can be extended through δ – i.e., T δ = τ – to an operator T : A(X ) → A such that ∥T ∥ = ∥τ ∥.
∗
Corresponding author. E-mail addresses: [email protected] (J.M.F. Castillo), [email protected] (J. Suárez).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.04.034
J.M.F. Castillo, J. Suárez / J. Math. Anal. Appl. 394 (2012) 152–158
153
The problem addressed and affirmatively solved in this paper is the following. Problem. Let A denote any of the following classes of L∞ -spaces: C (K )-spaces, Lindenstrauss spaces, λ-separably injective spaces, universally λ-separably injective spaces, λ-Lindenstrauss–Pełczyński spaces, L∞,λ -spaces. Does there exist an A-envelope of any given Banach space X ? To this end, we will develop a rather flexible device to construct such envelopes which unifies several constructions presented or outlined in the literature (cf., [6–11]). The final Remark in [7] can be understood as a hint that such unified approach was possible. The results in this paper solve the problems posed by the first author in his lecture during the Workshop in Topology and Banach spaces, celebrated in Łódź, Poland, 21–25 July 2010, and organized by P. Koszmider. It is a pleasure to thank Prof. Koszmider for the organization of such event and the invitation to participate. Moreover, the first author lectured about the results in this paper at the Banach space workshop celebrated at Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Alberta, Canada; from 4 to 9 March 2012. It is a pleasure to thank the organizers R. Anisca, S. Dilworth, E. Odell and B. Sari for the invitation to participate. Special thanks must go to the scientific director of BIRS, Prof. N. Ghoussoub and to BIRS Station Manager Brenda Williams for their exceptionally warm hospitality. The basic for our purposes is the push-out construction, which naturally appears when one considers a couple – or a family – of operators defined on the same space; in particular when considering the extension of an operator through an embedding. Let us explain why. Given operators α : Y → A and β : Y → B, the associated push-out diagram is α
Y −−−−→
β
A
′ β
(1)
α′
B −−−−→ PO . Here, the push-out space PO = PO(α, β) is the quotient of the direct sum A ⊕1 B, the product space endowed with the sum norm, by the closure of the subspace ∆ = {(α y, −β y) : y ∈ Y }. The map α ′ is given by the inclusion of B into A ⊕1 B followed by the natural quotient map A ⊕1 B → (A ⊕1 B)/∆, so that α ′ (b) = (0, b) + ∆ and, analogously, β ′ (a) = (a, 0) + ∆. The diagram (1) is commutative: β ′ α = α ′ β . Moreover, it is ‘minimal’ in the sense of having the following universal property: if β ′′ : A → C and α ′′ : B → C are operators such that β ′′ α = α ′′ β , then there is a unique operator γ : PO → C such that α ′′ = γ α ′ and β ′′ = γ β ′ . Clearly, γ ((a, b) + ∆) = β ′′ (a) + α ′′ (b) and one has ∥γ ∥ ≤ max{∥α ′′ ∥, ∥β ′′ ∥}. Regarding the behavior of the maps in diagram (1), one has [6] the following lemma. Lemma 1.2. (1) max{∥α ′ ∥, ∥β ′ ∥} ≤ 1. (2) If α is an isomorphic embedding, then ∆ is closed. (3) If α is an isometric embedding and ∥β∥ ≤ 1 then α ′ is an isometric embedding. (4) If α is an isomorphic embedding then α ′ is an isomorphic embedding. (5) If ∥β∥ ≤ 1 and α is an isomorphism then α ′ is an isomorphism and
∥(α ′ )−1 ∥ ≤ max{1, ∥α −1 ∥}. Returning to the using of the push-out construction, applying the previous situation when α is an isomorphic embedding and β an operator, we get that β ′ is an ‘‘extension’’ of β , except for the fact that one needs to increase the target space from B to PO. Nevertheless this increasing is as small and controlled as possible: indeed, it is not difficult to show [12] that PO /B and A/Y are isomorphic. The following result shows that it is enough to obtain envelopes of separable spaces. Let us recall that given an ordinal µ, a union ∪α<µ Xα is said to be continuous if for every limit ordinal β < µ, Xβ = ∪α<β Xα . Lemma 1.3. If separable spaces admit separable A-envelopes then all Banach spaces admit A-envelopes. Proof. We proceed by transfinite iteration on the density character of the space. For the case dens(X ) = ℵ0 , i.e., X separable, the existence of a separable A-envelope A(X ) is guaranteed by the hypothesis. Let κ be an ordinal, and assume that A-envelopes of spaces X with dens(X ) = ℵκ do exist. If dens(X ) = ℵκ+1 , let ωκ+1 be the first ordinal with cardinal |ωκ+1 | = ℵκ+1 . Then X can be represented as an increasing continuous union X =
α<ωκ+1
Xα
of spaces with dens(Xα ) ≤ ℵκ . The passage from α to α + 1 is described in the diagram Xα
δα
−−−−→ Xα+1 ′ δα
A(Xα ) −−−−→ POα −−−−→ A(POα ). ∆α
154
J.M.F. Castillo, J. Suárez / J. Math. Anal. Appl. 394 (2012) 152–158
Given an operator τ : X → A, its restriction τα : Xα → A admits an extension Tα : A(Xα ) → A with the same norm. This operator and τα+1 : Xα+1 → A plus the universal property of the push-out yield a simultaneous extension operator γ : POα → A with the same norm. This operator extends to an operator Tα+1 : A(POα ) → A with the same norm. Thus we can set A(Xα+1 ) = A(Pα ). Since dens POα = max{densXα+1 , densA(Xα )} we see that the relation densA(X ) = dens(X ) that holds for dens(X ) = ℵ0 is not spoiled through the inductive process. Thus, it makes sense to define
A(X ) =
α<ωκ+1
A(Xα ).
Let µ be a limit ordinal. Assume that the result has already been proved for spaces with density character ℵα with α < µ and we prove it for spaces X with dens(X ) = ℵµ , with µ a limit ordinal write µ = lim αj and then write X = ∪j Xαj – take the closure when it is a countable union – as a continuous increasing union of subspaces Xαj with densXαi = ℵαi . Then repeat the construction above following the diagram
−−−−→ Xαj δ′ αi
Xαi
δαi
A(Xαi ) −−−−→ POαi −−−−→ A(POαi ), ∆α
set A(Xαi+1 ) = A(POαi ) to finally put (take the closure when it is a countable union)
A(X ) =
α<µ
A(Xα ).
It is simple to see that the density of A(X ) is the same as that of X .
We show now that it is also enough to work with into isometries. Lemma 1.4. Let E , X be Banach spaces. Suppose there is a constant λ such that for each (isometric) subspace u : Y → E and every into isometry v : Y → X there exists an operator V : E → X such that Vu = v with ∥V ∥ ≤ λ. Then for every subspace Y of E every operator τ : Y → X can be extended to an operator T : E → X with ∥T ∥ ≤ λ∥τ ∥. Proof. Let τ : Y → X be a norm one operator. Denote by Y ′ the closure of the range of t and make the push-out of (u, t ): u
Y −−−−→
τ
E
′ t
(2)
u′
Y ′ −−−−→ PO . By Lemma 1.2, u′ is an into isometry, and the hypothesis yields an operator t ′′ : PO → X such that t ′′ u′ is the inclusion of Y ′ into X , with ∥t ′′ ∥ ≤ λ. Taking T = t ′′ t ′ the proof is done. 2. The device The input data we must introduce in the device are:
• • • • •
An ordinal µ. For each α < µ a Banach space Xα . For each α < µ a family Fα of Banach spaces. For each α < µ a family Jα of isometric embeddings between the spaces in Fα . For each α < µ a uniformly bounded family Lα of operators from the spaces in Fα to Xα+1 .
The output will be a Banach space Xµ plus an isometric embedding δ : X → Xµ with the property that ‘‘operators X → A of type Lα ’’ can be extended to an operator Xµ → A with the same norm. The relation between the class A and the input data is what transforms the meaningless ‘‘operators X → A of type Lα ’’ into ‘‘operators X → A whenever A ∈ A’’. To describe the device, observe that each φ ∈ Jα is an isometric embedding φ : j → k with j, k ∈ Fα . Let thus
Γα = {(φ, j, k) : φ : j → k; j, k ∈ Fα , φ ∈ Jα }. For each α < µ we consider the Banach spaces
ℓ1 (Γα , 1) = (x(φ,j,k) ) : x(φ,j,k) ∈ j, ∥x(φ,j,k) ∥ < +∞ ℓ1 (Γα , 2) = (x(φ,j,k) ) : x(φ,j,k) ∈ k, ∥x(φ,j,k) ∥ < +∞ .
J.M.F. Castillo, J. Suárez / J. Math. Anal. Appl. 394 (2012) 152–158
155
There is an obvious isometric embedding ⊕Jα : ℓ1 (Γα , 1) −→ ℓ1 (Γα , 2) defined by
(x(φ,j,k) ) −→ (φ x(φ,j,k) ). We also have an operator Σ Lα : ℓ1 (Γα , 1) −→ X1 given – with some abuse of notation to avoid three new indexes – by
(x(φ,j,k) ) −→
τ x(φ,j,k)
τ ∈Lα
whenever this makes sense. We start the device forming the push-out diagram ⊕J1
ℓ1 (Γ1 , 1) −−−−→ ℓ1 (Γ1 , 2) Σ L1 X1 −−−−→ PO2 . In this way we obtain an isometric enlargement ı : X1 → PO2 such that every operator t : j → X1 in L1 can be extended to an operator t ′ : k → PO2 through any embedding φ : j → k in J1 . In step α we proceed analogously forming the push-out diagram ⊕Jα
ℓ1 (Γα , 1) −−−−→ ℓ1 (Γα , 2) Σ Lα −−−−→ POα+1 .
Xα
In this way we obtain an isometric enlargement ıα : Xα → POα+1 such that every operator t : j → Xα in Lα can be extended to an operator t ′ : k → POα+1 through any embedding φ : j → k in Jα . Identifying Xα with POα allows to form the space Xβ = ∪α<β Xα for each limit ordinal β and thus the construction can be iterated until the ordinal µ. This provides the isometric embedding δ : X → Xµ . Assume now that given a class A the data Fα , Jα , Lα have been chosen so that the operators of Lα when acting from spaces of Fα into spaces of A admit equal norm extensions through the isometric embeddings of Jα . Given an operator τ : X → A with A ∈ A the composition τ Σ L1 admits an equal norm extension τ2 to X2 by the universal properties of the push-out. Repeating the same argument transfinitely many times we get an extension operator Xµ → A with the same norm. We show now how three classical constructions can be fit into this schema. 2.1. The Bourgain–Pisier construction Bourgain and Pisier [7] yield that every separable Banach space X can be, for every λ > 1, embedded into some
L∞,λ -space, L∞,λ (X ), in such a way that the corresponding quotient space L∞,λ (X )/X has the Schur and Radon–Nikodym properties. Let us briefly recall the construction. Assume that X = ∪Xn with each Xn finite dimensional, let in : Xn → X and a(1) in,n+1 : Xn → Xn+1 be the canonical inclusions. Fix λ > 1 and λ−1 < η < 1. Let s1 : S1 → l∞ be a subspace such that −1 there is an isomorphism u1 : S1 → X1 with ∥u1 ∥ ≤ η and ∥u1 ∥ ≤ λ. Form the push-out of s1 and i1 u1 to obtain a Banach a(1)
space E1 , an isometric embedding j1 : X → E1 and an embedding u1 : l∞ j1 i1 u1 = u1 s1 . s1
→ E1 making a commutative square, namely,
a(1)
S1 −−−−→ l∞
i1 u1
u 1 j1
X −−−−→ E1 . a(1)
We call PO1 the subspace of E1 that is the push-out of s1 and u1 . The space PO1 is λ-isomorphic to l∞ . Next, we form the a(1) push-out of the restriction of j1 to X1 and the inclusion X1 → X2 . This new push-out space is P1 = [j1 (X2 ) + u1 (l∞ )] (endowed with the norm of E1 ). a(2) For the next step, take s2 : S2 → l∞ a subspace such that there is an isomorphism u2 : S2 → P1 with ∥u2 ∥ ≤ η and 1 ∥ u− ∥ ≤ λ . Form the push-out of s and the composition S2 u2 : P1 → E1 that we call momentarily U2 . This yields a Banach 2 2 a(2)
space E2 with an isometric embedding j2 : E1 → E2 and an embedding u2 : l∞ → E2 making a commutative square, a(2) namely: j2 U2 = u2 s2 . We call PO2 the push-out of s2 and u2 , a subspace of E2 λ-isomorphic to l∞ . Form then the push-out of a(2) the restriction j2 : X2 → PO2 and the embedding X2 → X3 . This new push-out space is P2 = [j2 j1 (X3 ) + u2 (l∞ )] (endowed with the norm of E2 ). The process is iterated ω times so that the resulting L∞,λ (X ) superspace is the inductive limit j2 u1
j3 u2
PO1 −→ PO2 −→ PO3 −→ · · · while the embedding j : X → L∞,λ (X ) is given by j(x) = jn · · · j1 (x) when x ∈ Xn .
156
J.M.F. Castillo, J. Suárez / J. Math. Anal. Appl. 394 (2012) 152–158
This means that the space L∞,λ (X ) is the result of applying the device with the following input data:
• • • • •
The ordinal ω. For each n < ω the Banach space Xn .
a(n)
For each n < ω the family Fn is the couple Sn , ℓ∞ . For each n < ω the family Jn is the isometric embedding sn . 1 −1 For each n < +∞ the family Ln+1 is the operator in,n+1 un : Sn → Xn+1 . Here ∥un ∥ ≤ η and ∥u− < η < 1. n ∥ ≤ λ with λ
2.2. The Gurari˘ı and Kubis spaces These constructions appear described and studied in [6,13], to which we refer to for details. The Gurari˘ı space [8] is obtained setting the input data at
• The ordinal ω. • For each n < ω the Banach space Xn = POn and X1 is any separable Banach space. • For each n < ω, Fn is a countable family of finite-dimensional Banach spaces that is dense, in the Banach–Mazur distance, in the space of all finite-dimensional Banach spaces.
• For each n < ω, Jn denotes a countable and dense family of almost isometric embeddings between elements of Fn . • For each n < +∞, Ln+1 is the family of all isometric embeddings of elements of Fn into Xn . The resulting space G = Xω – see [6] – is the only (up to isometries) separable Banach space with the following property called ‘‘almost universal disposition for finite-dimensional spaces’’: given an isometric embedding between two finite dimensional spaces F → G and ε > 0, every into isometry F → G can be extended to a (1 + ε)-isometry G → G. The Kubis space [14] is obtained setting the input data at
• • • • •
The ordinal ω1 . For each countable α Xα = POα and X1 is any separable Banach space. For each countable α , Fα is the family of all separable Banach spaces. For each countable α , Jα denotes the family of all isometric embeddings between elements of Fα . For each countable α , Lα+1 is the family of all isometric embeddings of elements of Fα into Xα .
The resulting space K = Xω1 – see [6] – is, under CH, the only (up to isometries) space having density character ℵ1 with the following property, called ‘‘universal disposition for separable spaces’’: given an isometric embedding between two separable spaces S → S ′ , every into isometry S → K can be extended to an into isometry S ′ → K . 3. The L∞,λ -envelope We proceed as in the Bourgain–Pisier construction, except that we set η = λ−1 , which requires the further modification of embedding each Sn isometrically into C [0, 1] since there is no margin to embed them isometrically into any ℓn∞ . Thus, assuming X is separable, the input data are
• • • • •
The ordinal ω. For each n < ω the Banach space Xn . For each n < ω the family Fn is the couple Sn , C [0, 1]. For each n < ω the family Jn is an isometric embedding sn : Sn → C [0, 1]. 1 For each n < +∞ the family Ln+1 is the operator in,n+1 un : Sn → Xn+1 . Here ∥un ∥ ≤ λ−1 and ∥u− n ∥ ≤ λ. The outcome space Xω is an L∞,λ since it is the inductive limit of the spaces POn appearing in the diagram sn
Sn −−−−→ C [0, 1]
un
u n n1
Xn −−−−→ POn+1 which therefore must be λ-isomorphic copies of C [0, 1]. The general nonseparable case follows from Lemma 1.3. Now, let τ : X → E be a norm one operator from X into an L∞,λ -space. Each composition τ un is a finite dimensional operator with norm at most λ−1 . It therefore admits a norm one extension Tn : C [0, 1] → E. The universal property of the push-out provides a further norm one extension to PO1 , and thus the extension continues inductively up to Xω . One therefore has the following theorem. Theorem 3.1. Let λ > 1. Every Banach space X can be isometrically embedded into an L∞,λ -space Lλ (X ) with the property that every operator τ : X → E from X into an L∞,λ space E can be extended to an operator T : Lλ (X ) → E with ∥T ∥ = ∥τ ∥.
J.M.F. Castillo, J. Suárez / J. Math. Anal. Appl. 394 (2012) 152–158
157
Two closely related results appeared in [11]: Thm 2.1 asserts that for a separable X , λ > 1 and ε > 0 one can isometrically embed X into some L∞,λ -space L∞,λ (X ) in such a way that every operator τ : X → E from X into an L∞,λ -space E admits an extension T : L∞,λ (X ) → E with the estimate ∥T ∥ ≤ (1 + ε)∥τ ∥ and, moreover, the quotient L∞,λ (X )/X has the Radon–Nikodym and the Schur properties; while Thm 2.2 isometrically embeds a separable X into some L∞,λ -space L∞,λ (X ) in such a way that every operator τ : X → E from X into an L∞,µ -space E, with µ < λ, admits an extension T : L∞,λ (X ) → E with the same norm. Those constructions do not yet provide L∞,λ -envelopes because of the ε , in the first case, and the condition µ < λ in the second. The construction of L∞,λ -envelopes is especially interesting since these seem to be the first results about the extension of L∞ -valued operators. Observe that in our construction we have lost the property of making Schur the quotient space L∞,λ (X )/X , since this property was a direct consequence of the choice λ−1 < η < 1. It is a tantalizing conjecture that the Bourgain–Pisier construction may have the isometric extension property with respect to L∞,λ -valued operators. In [9] it was established that it enjoys the property of admitting almost isometric extensions for Lindenstrauss-valued operators. For the nonseparable version of the Bourgain–Pisier construction obtained by Abad and Todorcevic [15] no extension property is known. 4. The Lindenstrauss and the C (K )-envelope Proposition 4.1 in [9] asserts that every separable Banach space X can be isometrically embedded into some Lindenstrauss space L (X ) in such a way that every operator τ : X → E from X into a Lindenstrauss space E admits, for every ε > 0, an extension T : L (X ) → E with an estimate ∥T ∥ ≤ (1 + ε)∥τ ∥. The assertion of that same theorem for nonseparable spaces is not justified. To construct the Lindenstrauss envelope of a separable Banach space we set the input data of the device – with the same notation as in the Bourgain–Pisier construction – as follows:
• • • • •
The ordinal ω. For each n < ω the Banach space Xn . a(n) For each n < ω the family Fn is the couple Sn , ℓ∞ . a(n) For each n < ω the family Jn is the isometric embedding sn : Sn → ℓ∞ . 1 For each n < +∞ the family Ln+1 is the operator in,n+1 un : Sn → Xn+1 . Here ∥un ∥ ≤ ηn and ∥u− n ∥ ≤ λn with −1 λn < ηn < 1 and lim λn = 1.
We prove the result for separable X , while the general case follows from Lemma 1.3. The outcome space Xω is an L∞,1+ space since it is the inductive limit of the spaces POn appearing in the diagram sn
Sn −−−−→
un
ℓa∞(n) u n
n1
Xn −−−−→ POn+1 , a(n)
which are λn -isomorphic copies of ℓ∞ . Let τ : X → E be a norm one operator from X into a Lindenstrauss space E. Each composition τ un is a finite dimensional operator with norm at most ηn . It therefore admits, for every ε > 0, an extension a(n) Tn : ℓ∞ → E with norm at most (1 + ε)ηn . The universal property of the push-out provides a further extension to PO1 with norm (1 + ε)ηn . Choose ε ≤ ηn−1 to get a norm one extension to PO1 . Then continue this procedure inductively up to get a norm one operator on Xω . One therefore has the following theorem. Theorem 4.1. Every Banach space X can be isometrically embedded into a Lindenstrauss space L (X ) with the property that every operator τ : X → E from X into a Lindenstrauss space E can be extended to an operator T : L (X ) → E such that ∥T ∥ = ∥τ ∥. Since every separable Lindenstrauss space is 1-complemented in the Gurari˘ı space, it turns out that the Gurari˘ı space (with some suitable isometric embedding) provides a Lindenstrauss envelope for every separable Banach space. As for the C (K )-envelope, we have already remarked that the canonical isometric embedding X → C (BX ∗ ) defines a C (K )-envelope for X . Milutin’s theorem yields that for separable X the spaces C (BX ∗ ) and C [0, 1] are isomorphic. They are not isometric, however, and therefore one still cannot guarantee that C [0, 1] provides a C (K )-envelope for every separable Banach space (through some suitable isometric embedding). Nevertheless, this is true since [16] every separable C (K ) can be isometrically embedded as a 1-complemented subspace of C [0, 1]. 5. The E-envelope of X We present a general construction from which the (universally) separably injective or Lindenstrauss–Pełczyński envelope can be easily derived. Let E , X be Banach spaces. Let µ be the first ordinal whose associated cardinal has cofinality greater than densE. Set the input data of the device as:
• The ordinal µ. • For each α < µ, Xα = POα and X1 = X .
158
J.M.F. Castillo, J. Suárez / J. Math. Anal. Appl. 394 (2012) 152–158
• For each α < µ, Fα is the family of all subspaces of E. • For each α < µ, Jα denotes the family of all isometric embeddings between elements of Fα . • For each α < µ, Lα+1 is the family of all isometric embeddings of elements of Fα into Xα . Let us call the final space Xµ = Xµ (E ). This space deserves the name of E-envelope of X due to the following properties it enjoys. Proposition 5.1. Let E , X be Banach spaces. (1) For every subspace i : Y → E every operator τ : Y → Xµ (E ) can be extended to an operator T : E → Xµ with ∥T ∥ = ∥τ ∥. (2) Assume that Z has the property that for every subspace i : Y → E every operator τ : Y → Z can be extended to an operator T : E → Z with ∥T ∥ = ∥τ ∥. Then every operator τ : X → Z can be extended to an operator T : Xµ (X ) → Z with ∥T ∥ = ∥τ ∥. Proof. Assertion (1) holds by the assumption on the cofinality of µ: for every subspace Y of E and every operator τ the image τ (E ) must lie in some Xα with α < µ. Thus, every into isometry Y → Xµ is actually an into isometry Y → Xµ , being thus one of the elements of Jα which can therefore be extended to an into isometry E → POα+1 , hence to an into isometry X → Xµ through the embedding Y → E. Lemma 1.4 yields the extension of arbitrary operators. Assertion (2) is clear by transfinite induction, whose step α is as follows: the composition τ Σ Lα extends through ⊕Jα without increasing the norm. The universal property of the push out yields a simultaneous extension to Xα+1 = POα+1 . The choices E = ℓ1 , ℓ∞ , c0 and the remarks at the introduction immediately yield the following.
• The space Xω1 (ℓ1 ) is a 1-separably injective envelope of X . • The space Xω1 (ℓ∞ ) is a universally 1-separably injective envelope of X . • The space Xω1 (c0 ) is a 1-Lindenstrauss–Pełczyński envelope of X . Given λ ≥ 1, replacing Lα by the family λ−1 Lα = {λ−1 τ : τ ∈ Lα } one obtains the respective λ- envelopes. One therefore has the following theorem. Theorem 5.2. For every Banach space X and every λ ≥ 1 there exists the following.
• An isometric embedding of X into a λ-separably injective space Θ (X ) in such a way that every operator τ : X → Z from X into a λ-separably injective space Z admits an extension T : Θ (X ) → Z with ∥T ∥ = ∥τ ∥. • An isometric embedding of X into a universally λ-separably injective space U(X ) in such a way that every operator τ : X → Z from X into a universally λ-separably injective space Z admits an extension T : U(X ) → Z with ∥T ∥ = ∥τ ∥. • An isometric embedding of X into a λ-Lindenstrauss–Pełczyński space L P (X ) in such a way that every operator τ : X → Z from X into a λ-Lindenstrauss–Pełczyński space Z admits an extension T : L P (X ) → Z with ∥T ∥ = ∥τ ∥. Observe that for separable X any isometric embedding X → ℓ∞ works as a universally 1-separably injective envelope of X . Acknowledgments This research has been supported in part by project MTM2010-20190-C02-01 and the program Junta de Extremadura GR10113 IV Plan Regional I + D + i, Ayudas a Grupos de Investigación. References [1] M. Zippin, Extension of bounded linear operators, in: W.B. Johnson, J. Lindenstrauss (Eds.), Handbook of the Geometry of Banach Spaces, vol. 2, Elsevier, 2003, pp. 1703–1741. [2] J.M.F. Castillo, The hitchhiker guide to categorical Banach space theory. Part I, Extracta Math. 25 (2010) 103–149. [3] A. Avilés, F. Cabello, J.M.F. Castillo, M. González, Y. Moreno, On separably injective Banach spaces (2011), preprint arXiv:1103.6064. [4] J.M.F. Castillo, Y. Moreno, J. Suárez, On Lindenstrauss–Pelczyński spaces, Studia Math. 174 (2006) 213–231. [5] J.M.F. Castillo, Y. Moreno, J. Suárez, On the structure of Lindenstrauss–Pelczyński spaces, Studia Math. 194 (2009) 105–115. [6] A. Avilés, F. Cabello, J.M.F. Castillo, M. González, Y. Moreno, On Banach spaces of universal disposition, J. Funct. Anal. 261 (2011) 2347–2361. [7] J. Bourgain, G. Pisier, A construction of L∞ -spaces and related Banach spaces, Bol. Soc. Bras. Mat. 14 (1983) 109–123. [8] V.I. Gurari˘ı, Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces, Sibirsk. Mat. Ž 7 (1966) 1002–1013 (in Russian). [9] J.M.F. Castillo, J. Suárez, Extending operators into Lindenstrauss spaces, Israel J. Math. 169 (2009) 1–27. [10] G. Pisier, Counterexample to a conjecture of Grothendieck, Acta Math. 151 (1983) 181–208. [11] J. Suárez de la Fuente, An extension property of the Bourgain–Pisier construction. Ark. Math. (in press) http://dx.doi.org/10.1007/s11512-012-0166-8. [12] J.M.F. Castillo, M. González, Three-space problems in Banach space theory, in: Lecture Notes in Math., vol. 1667, Springer-Verlag, 1997. [13] J. Garbulinska, W. Kubis, Remarks on Gurari˘ı spaces, Extracta Math. 25 (2011). [14] W. Kubis, Fraïssé sequences — a category-theoretic approach to universal homogeneous structures (2007), arXiv:0711.1683v1. [15] J. López-Abad, S. Todorcevic, personal communication. [16] H.P. Rosenthal, The Banach spaces C (K ), in: W.B. Johnson, J. Lindenstrauss (Eds.), Handbook of the Geometry of Banach Spaces, vol. 2, Elsevier, 2003, pp. 1547–1602.
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
On L∞ -envelopes of Banach spaces Jesús M.F. Castillo a,∗ , Jesús Suárez b a
Departamento de Matemáticas, Facultad de Ciencias, Univ. de Extremadura, Avda. de Elvas s/n, 06071 Badajoz, Spain
b
Escuela Politécnica, Universidad de Extremadura, Avda. de la Universidad s/n, 10071 Cáceres, Spain
article
info
Article history: Received 24 June 2011 Available online 28 April 2012 Submitted by R.M. Aron Keywords: Extension of operators L∞ -spaces
abstract Let A denote any of the following classes of L∞ -spaces: C (K )-spaces, Lindenstrauss spaces, λ-separably injective spaces, universally λ-separably injective spaces, λ-Lindenstrauss– Pełczyński spaces or L∞,λ -spaces. We show that every Banach space X can be isometrically embedded into a space A(X ) ∈ A so that every operator X → A with A ∈ A can be extended to an operator A(X ) → A with the same norm. © 2012 Elsevier Inc. All rights reserved.
1. Introduction and preliminaries As is well-known, every Banach space X can be naturally embedded into the space of continuous functions C (BX ∗ ) in such a way that the embedding δX : X −→ C (BX ∗ ) enjoys the following universal property: Every C (K ) − valued operator τ : X → C (K ) can be extended through δX to an operator T : C (BX ∗ ) → C (K ) so that ∥T ∥ = ∥τ ∥. See [1], and also [2] for a historical account and a categorical presentation of this result. Does there exist a similar universal embedding for other classes of Banach spaces? It is particularly interesting to consider the situation replacing C (K )-spaces by their ‘‘local’’ versions, the L∞ spaces. Recall that a Banach space X is said to be an L∞,λ -space, λ ≥ 1, if every finite dimensional subspace F of X is contained in another finite dimensional subspace G of X whose Banach–Mazur distance to the corresponding ℓdimG is at most λ. A space X is said to be an L∞ -space if it is an L∞,λ -space for some λ ≥ 1. The ∞ following subclasses of L∞ -spaces have appeared in the literature (the missing definitions can be found in the next paragraph): L∞,λ -spaces, λ-Lindenstrauss–Pełczyński spaces, Lindenstrauss spaces, C (K )-spaces, λ-separably injective spaces, universally λ-separably injective spaces and λ-injective spaces. A Banach space is said to be a Lindenstrauss space if it is an L∞,1+ε -space for all ε > 0. The space X is said to be injective if every operator τ : Y → X can be extended to an operator T : W → X for any bigger superspace W of Y . The space X is said to be separably injective (resp. universally separably injective) [3] when the condition above is satisfied for separable W (resp. separable) Y . It is not hard to see [3] that for separable injectivity it is enough to take W = ℓ1 ; and W = ℓ∞ for universal separable injectivity. The space X is said to be a Lindenstrauss–Pełczyński space [4,5] when the previous condition holds for W = c0 . When an extension T verifying ∥T ∥ ≤ λ∥τ ∥ exists then the classes will be referred to, respectively, as λ-injective, λ-separably injective, universally λ-separably injective, and λ-Lindenstrauss–Pełczyński spaces. Definition 1.1. Let A be a class of Banach spaces. An A-envelope of a Banach space X is a couple (A(X ), δ) formed by a Banach space A(X ) ∈ A and an isometric embedding δ : X → A(X ) with the property that for every A ∈ A every operator τ : X → A can be extended through δ – i.e., T δ = τ – to an operator T : A(X ) → A such that ∥T ∥ = ∥τ ∥.
∗
Corresponding author. E-mail addresses: [email protected] (J.M.F. Castillo), [email protected] (J. Suárez).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.04.034
J.M.F. Castillo, J. Suárez / J. Math. Anal. Appl. 394 (2012) 152–158
153
The problem addressed and affirmatively solved in this paper is the following. Problem. Let A denote any of the following classes of L∞ -spaces: C (K )-spaces, Lindenstrauss spaces, λ-separably injective spaces, universally λ-separably injective spaces, λ-Lindenstrauss–Pełczyński spaces, L∞,λ -spaces. Does there exist an A-envelope of any given Banach space X ? To this end, we will develop a rather flexible device to construct such envelopes which unifies several constructions presented or outlined in the literature (cf., [6–11]). The final Remark in [7] can be understood as a hint that such unified approach was possible. The results in this paper solve the problems posed by the first author in his lecture during the Workshop in Topology and Banach spaces, celebrated in Łódź, Poland, 21–25 July 2010, and organized by P. Koszmider. It is a pleasure to thank Prof. Koszmider for the organization of such event and the invitation to participate. Moreover, the first author lectured about the results in this paper at the Banach space workshop celebrated at Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Alberta, Canada; from 4 to 9 March 2012. It is a pleasure to thank the organizers R. Anisca, S. Dilworth, E. Odell and B. Sari for the invitation to participate. Special thanks must go to the scientific director of BIRS, Prof. N. Ghoussoub and to BIRS Station Manager Brenda Williams for their exceptionally warm hospitality. The basic for our purposes is the push-out construction, which naturally appears when one considers a couple – or a family – of operators defined on the same space; in particular when considering the extension of an operator through an embedding. Let us explain why. Given operators α : Y → A and β : Y → B, the associated push-out diagram is α
Y −−−−→
β
A
′ β
(1)
α′
B −−−−→ PO . Here, the push-out space PO = PO(α, β) is the quotient of the direct sum A ⊕1 B, the product space endowed with the sum norm, by the closure of the subspace ∆ = {(α y, −β y) : y ∈ Y }. The map α ′ is given by the inclusion of B into A ⊕1 B followed by the natural quotient map A ⊕1 B → (A ⊕1 B)/∆, so that α ′ (b) = (0, b) + ∆ and, analogously, β ′ (a) = (a, 0) + ∆. The diagram (1) is commutative: β ′ α = α ′ β . Moreover, it is ‘minimal’ in the sense of having the following universal property: if β ′′ : A → C and α ′′ : B → C are operators such that β ′′ α = α ′′ β , then there is a unique operator γ : PO → C such that α ′′ = γ α ′ and β ′′ = γ β ′ . Clearly, γ ((a, b) + ∆) = β ′′ (a) + α ′′ (b) and one has ∥γ ∥ ≤ max{∥α ′′ ∥, ∥β ′′ ∥}. Regarding the behavior of the maps in diagram (1), one has [6] the following lemma. Lemma 1.2. (1) max{∥α ′ ∥, ∥β ′ ∥} ≤ 1. (2) If α is an isomorphic embedding, then ∆ is closed. (3) If α is an isometric embedding and ∥β∥ ≤ 1 then α ′ is an isometric embedding. (4) If α is an isomorphic embedding then α ′ is an isomorphic embedding. (5) If ∥β∥ ≤ 1 and α is an isomorphism then α ′ is an isomorphism and
∥(α ′ )−1 ∥ ≤ max{1, ∥α −1 ∥}. Returning to the using of the push-out construction, applying the previous situation when α is an isomorphic embedding and β an operator, we get that β ′ is an ‘‘extension’’ of β , except for the fact that one needs to increase the target space from B to PO. Nevertheless this increasing is as small and controlled as possible: indeed, it is not difficult to show [12] that PO /B and A/Y are isomorphic. The following result shows that it is enough to obtain envelopes of separable spaces. Let us recall that given an ordinal µ, a union ∪α<µ Xα is said to be continuous if for every limit ordinal β < µ, Xβ = ∪α<β Xα . Lemma 1.3. If separable spaces admit separable A-envelopes then all Banach spaces admit A-envelopes. Proof. We proceed by transfinite iteration on the density character of the space. For the case dens(X ) = ℵ0 , i.e., X separable, the existence of a separable A-envelope A(X ) is guaranteed by the hypothesis. Let κ be an ordinal, and assume that A-envelopes of spaces X with dens(X ) = ℵκ do exist. If dens(X ) = ℵκ+1 , let ωκ+1 be the first ordinal with cardinal |ωκ+1 | = ℵκ+1 . Then X can be represented as an increasing continuous union X =
α<ωκ+1
Xα
of spaces with dens(Xα ) ≤ ℵκ . The passage from α to α + 1 is described in the diagram Xα
δα
−−−−→ Xα+1 ′ δα
A(Xα ) −−−−→ POα −−−−→ A(POα ). ∆α
154
J.M.F. Castillo, J. Suárez / J. Math. Anal. Appl. 394 (2012) 152–158
Given an operator τ : X → A, its restriction τα : Xα → A admits an extension Tα : A(Xα ) → A with the same norm. This operator and τα+1 : Xα+1 → A plus the universal property of the push-out yield a simultaneous extension operator γ : POα → A with the same norm. This operator extends to an operator Tα+1 : A(POα ) → A with the same norm. Thus we can set A(Xα+1 ) = A(Pα ). Since dens POα = max{densXα+1 , densA(Xα )} we see that the relation densA(X ) = dens(X ) that holds for dens(X ) = ℵ0 is not spoiled through the inductive process. Thus, it makes sense to define
A(X ) =
α<ωκ+1
A(Xα ).
Let µ be a limit ordinal. Assume that the result has already been proved for spaces with density character ℵα with α < µ and we prove it for spaces X with dens(X ) = ℵµ , with µ a limit ordinal write µ = lim αj and then write X = ∪j Xαj – take the closure when it is a countable union – as a continuous increasing union of subspaces Xαj with densXαi = ℵαi . Then repeat the construction above following the diagram
−−−−→ Xαj δ′ αi
Xαi
δαi
A(Xαi ) −−−−→ POαi −−−−→ A(POαi ), ∆α
set A(Xαi+1 ) = A(POαi ) to finally put (take the closure when it is a countable union)
A(X ) =
α<µ
A(Xα ).
It is simple to see that the density of A(X ) is the same as that of X .
We show now that it is also enough to work with into isometries. Lemma 1.4. Let E , X be Banach spaces. Suppose there is a constant λ such that for each (isometric) subspace u : Y → E and every into isometry v : Y → X there exists an operator V : E → X such that Vu = v with ∥V ∥ ≤ λ. Then for every subspace Y of E every operator τ : Y → X can be extended to an operator T : E → X with ∥T ∥ ≤ λ∥τ ∥. Proof. Let τ : Y → X be a norm one operator. Denote by Y ′ the closure of the range of t and make the push-out of (u, t ): u
Y −−−−→
τ
E
′ t
(2)
u′
Y ′ −−−−→ PO . By Lemma 1.2, u′ is an into isometry, and the hypothesis yields an operator t ′′ : PO → X such that t ′′ u′ is the inclusion of Y ′ into X , with ∥t ′′ ∥ ≤ λ. Taking T = t ′′ t ′ the proof is done. 2. The device The input data we must introduce in the device are:
• • • • •
An ordinal µ. For each α < µ a Banach space Xα . For each α < µ a family Fα of Banach spaces. For each α < µ a family Jα of isometric embeddings between the spaces in Fα . For each α < µ a uniformly bounded family Lα of operators from the spaces in Fα to Xα+1 .
The output will be a Banach space Xµ plus an isometric embedding δ : X → Xµ with the property that ‘‘operators X → A of type Lα ’’ can be extended to an operator Xµ → A with the same norm. The relation between the class A and the input data is what transforms the meaningless ‘‘operators X → A of type Lα ’’ into ‘‘operators X → A whenever A ∈ A’’. To describe the device, observe that each φ ∈ Jα is an isometric embedding φ : j → k with j, k ∈ Fα . Let thus
Γα = {(φ, j, k) : φ : j → k; j, k ∈ Fα , φ ∈ Jα }. For each α < µ we consider the Banach spaces
ℓ1 (Γα , 1) = (x(φ,j,k) ) : x(φ,j,k) ∈ j, ∥x(φ,j,k) ∥ < +∞ ℓ1 (Γα , 2) = (x(φ,j,k) ) : x(φ,j,k) ∈ k, ∥x(φ,j,k) ∥ < +∞ .
J.M.F. Castillo, J. Suárez / J. Math. Anal. Appl. 394 (2012) 152–158
155
There is an obvious isometric embedding ⊕Jα : ℓ1 (Γα , 1) −→ ℓ1 (Γα , 2) defined by
(x(φ,j,k) ) −→ (φ x(φ,j,k) ). We also have an operator Σ Lα : ℓ1 (Γα , 1) −→ X1 given – with some abuse of notation to avoid three new indexes – by
(x(φ,j,k) ) −→
τ x(φ,j,k)
τ ∈Lα
whenever this makes sense. We start the device forming the push-out diagram ⊕J1
ℓ1 (Γ1 , 1) −−−−→ ℓ1 (Γ1 , 2) Σ L1 X1 −−−−→ PO2 . In this way we obtain an isometric enlargement ı : X1 → PO2 such that every operator t : j → X1 in L1 can be extended to an operator t ′ : k → PO2 through any embedding φ : j → k in J1 . In step α we proceed analogously forming the push-out diagram ⊕Jα
ℓ1 (Γα , 1) −−−−→ ℓ1 (Γα , 2) Σ Lα −−−−→ POα+1 .
Xα
In this way we obtain an isometric enlargement ıα : Xα → POα+1 such that every operator t : j → Xα in Lα can be extended to an operator t ′ : k → POα+1 through any embedding φ : j → k in Jα . Identifying Xα with POα allows to form the space Xβ = ∪α<β Xα for each limit ordinal β and thus the construction can be iterated until the ordinal µ. This provides the isometric embedding δ : X → Xµ . Assume now that given a class A the data Fα , Jα , Lα have been chosen so that the operators of Lα when acting from spaces of Fα into spaces of A admit equal norm extensions through the isometric embeddings of Jα . Given an operator τ : X → A with A ∈ A the composition τ Σ L1 admits an equal norm extension τ2 to X2 by the universal properties of the push-out. Repeating the same argument transfinitely many times we get an extension operator Xµ → A with the same norm. We show now how three classical constructions can be fit into this schema. 2.1. The Bourgain–Pisier construction Bourgain and Pisier [7] yield that every separable Banach space X can be, for every λ > 1, embedded into some
L∞,λ -space, L∞,λ (X ), in such a way that the corresponding quotient space L∞,λ (X )/X has the Schur and Radon–Nikodym properties. Let us briefly recall the construction. Assume that X = ∪Xn with each Xn finite dimensional, let in : Xn → X and a(1) in,n+1 : Xn → Xn+1 be the canonical inclusions. Fix λ > 1 and λ−1 < η < 1. Let s1 : S1 → l∞ be a subspace such that −1 there is an isomorphism u1 : S1 → X1 with ∥u1 ∥ ≤ η and ∥u1 ∥ ≤ λ. Form the push-out of s1 and i1 u1 to obtain a Banach a(1)
space E1 , an isometric embedding j1 : X → E1 and an embedding u1 : l∞ j1 i1 u1 = u1 s1 . s1
→ E1 making a commutative square, namely,
a(1)
S1 −−−−→ l∞
i1 u1
u 1 j1
X −−−−→ E1 . a(1)
We call PO1 the subspace of E1 that is the push-out of s1 and u1 . The space PO1 is λ-isomorphic to l∞ . Next, we form the a(1) push-out of the restriction of j1 to X1 and the inclusion X1 → X2 . This new push-out space is P1 = [j1 (X2 ) + u1 (l∞ )] (endowed with the norm of E1 ). a(2) For the next step, take s2 : S2 → l∞ a subspace such that there is an isomorphism u2 : S2 → P1 with ∥u2 ∥ ≤ η and 1 ∥ u− ∥ ≤ λ . Form the push-out of s and the composition S2 u2 : P1 → E1 that we call momentarily U2 . This yields a Banach 2 2 a(2)
space E2 with an isometric embedding j2 : E1 → E2 and an embedding u2 : l∞ → E2 making a commutative square, a(2) namely: j2 U2 = u2 s2 . We call PO2 the push-out of s2 and u2 , a subspace of E2 λ-isomorphic to l∞ . Form then the push-out of a(2) the restriction j2 : X2 → PO2 and the embedding X2 → X3 . This new push-out space is P2 = [j2 j1 (X3 ) + u2 (l∞ )] (endowed with the norm of E2 ). The process is iterated ω times so that the resulting L∞,λ (X ) superspace is the inductive limit j2 u1
j3 u2
PO1 −→ PO2 −→ PO3 −→ · · · while the embedding j : X → L∞,λ (X ) is given by j(x) = jn · · · j1 (x) when x ∈ Xn .
156
J.M.F. Castillo, J. Suárez / J. Math. Anal. Appl. 394 (2012) 152–158
This means that the space L∞,λ (X ) is the result of applying the device with the following input data:
• • • • •
The ordinal ω. For each n < ω the Banach space Xn .
a(n)
For each n < ω the family Fn is the couple Sn , ℓ∞ . For each n < ω the family Jn is the isometric embedding sn . 1 −1 For each n < +∞ the family Ln+1 is the operator in,n+1 un : Sn → Xn+1 . Here ∥un ∥ ≤ η and ∥u− < η < 1. n ∥ ≤ λ with λ
2.2. The Gurari˘ı and Kubis spaces These constructions appear described and studied in [6,13], to which we refer to for details. The Gurari˘ı space [8] is obtained setting the input data at
• The ordinal ω. • For each n < ω the Banach space Xn = POn and X1 is any separable Banach space. • For each n < ω, Fn is a countable family of finite-dimensional Banach spaces that is dense, in the Banach–Mazur distance, in the space of all finite-dimensional Banach spaces.
• For each n < ω, Jn denotes a countable and dense family of almost isometric embeddings between elements of Fn . • For each n < +∞, Ln+1 is the family of all isometric embeddings of elements of Fn into Xn . The resulting space G = Xω – see [6] – is the only (up to isometries) separable Banach space with the following property called ‘‘almost universal disposition for finite-dimensional spaces’’: given an isometric embedding between two finite dimensional spaces F → G and ε > 0, every into isometry F → G can be extended to a (1 + ε)-isometry G → G. The Kubis space [14] is obtained setting the input data at
• • • • •
The ordinal ω1 . For each countable α Xα = POα and X1 is any separable Banach space. For each countable α , Fα is the family of all separable Banach spaces. For each countable α , Jα denotes the family of all isometric embeddings between elements of Fα . For each countable α , Lα+1 is the family of all isometric embeddings of elements of Fα into Xα .
The resulting space K = Xω1 – see [6] – is, under CH, the only (up to isometries) space having density character ℵ1 with the following property, called ‘‘universal disposition for separable spaces’’: given an isometric embedding between two separable spaces S → S ′ , every into isometry S → K can be extended to an into isometry S ′ → K . 3. The L∞,λ -envelope We proceed as in the Bourgain–Pisier construction, except that we set η = λ−1 , which requires the further modification of embedding each Sn isometrically into C [0, 1] since there is no margin to embed them isometrically into any ℓn∞ . Thus, assuming X is separable, the input data are
• • • • •
The ordinal ω. For each n < ω the Banach space Xn . For each n < ω the family Fn is the couple Sn , C [0, 1]. For each n < ω the family Jn is an isometric embedding sn : Sn → C [0, 1]. 1 For each n < +∞ the family Ln+1 is the operator in,n+1 un : Sn → Xn+1 . Here ∥un ∥ ≤ λ−1 and ∥u− n ∥ ≤ λ. The outcome space Xω is an L∞,λ since it is the inductive limit of the spaces POn appearing in the diagram sn
Sn −−−−→ C [0, 1]
un
u n n1
Xn −−−−→ POn+1 which therefore must be λ-isomorphic copies of C [0, 1]. The general nonseparable case follows from Lemma 1.3. Now, let τ : X → E be a norm one operator from X into an L∞,λ -space. Each composition τ un is a finite dimensional operator with norm at most λ−1 . It therefore admits a norm one extension Tn : C [0, 1] → E. The universal property of the push-out provides a further norm one extension to PO1 , and thus the extension continues inductively up to Xω . One therefore has the following theorem. Theorem 3.1. Let λ > 1. Every Banach space X can be isometrically embedded into an L∞,λ -space Lλ (X ) with the property that every operator τ : X → E from X into an L∞,λ space E can be extended to an operator T : Lλ (X ) → E with ∥T ∥ = ∥τ ∥.
J.M.F. Castillo, J. Suárez / J. Math. Anal. Appl. 394 (2012) 152–158
157
Two closely related results appeared in [11]: Thm 2.1 asserts that for a separable X , λ > 1 and ε > 0 one can isometrically embed X into some L∞,λ -space L∞,λ (X ) in such a way that every operator τ : X → E from X into an L∞,λ -space E admits an extension T : L∞,λ (X ) → E with the estimate ∥T ∥ ≤ (1 + ε)∥τ ∥ and, moreover, the quotient L∞,λ (X )/X has the Radon–Nikodym and the Schur properties; while Thm 2.2 isometrically embeds a separable X into some L∞,λ -space L∞,λ (X ) in such a way that every operator τ : X → E from X into an L∞,µ -space E, with µ < λ, admits an extension T : L∞,λ (X ) → E with the same norm. Those constructions do not yet provide L∞,λ -envelopes because of the ε , in the first case, and the condition µ < λ in the second. The construction of L∞,λ -envelopes is especially interesting since these seem to be the first results about the extension of L∞ -valued operators. Observe that in our construction we have lost the property of making Schur the quotient space L∞,λ (X )/X , since this property was a direct consequence of the choice λ−1 < η < 1. It is a tantalizing conjecture that the Bourgain–Pisier construction may have the isometric extension property with respect to L∞,λ -valued operators. In [9] it was established that it enjoys the property of admitting almost isometric extensions for Lindenstrauss-valued operators. For the nonseparable version of the Bourgain–Pisier construction obtained by Abad and Todorcevic [15] no extension property is known. 4. The Lindenstrauss and the C (K )-envelope Proposition 4.1 in [9] asserts that every separable Banach space X can be isometrically embedded into some Lindenstrauss space L (X ) in such a way that every operator τ : X → E from X into a Lindenstrauss space E admits, for every ε > 0, an extension T : L (X ) → E with an estimate ∥T ∥ ≤ (1 + ε)∥τ ∥. The assertion of that same theorem for nonseparable spaces is not justified. To construct the Lindenstrauss envelope of a separable Banach space we set the input data of the device – with the same notation as in the Bourgain–Pisier construction – as follows:
• • • • •
The ordinal ω. For each n < ω the Banach space Xn . a(n) For each n < ω the family Fn is the couple Sn , ℓ∞ . a(n) For each n < ω the family Jn is the isometric embedding sn : Sn → ℓ∞ . 1 For each n < +∞ the family Ln+1 is the operator in,n+1 un : Sn → Xn+1 . Here ∥un ∥ ≤ ηn and ∥u− n ∥ ≤ λn with −1 λn < ηn < 1 and lim λn = 1.
We prove the result for separable X , while the general case follows from Lemma 1.3. The outcome space Xω is an L∞,1+ space since it is the inductive limit of the spaces POn appearing in the diagram sn
Sn −−−−→
un
ℓa∞(n) u n
n1
Xn −−−−→ POn+1 , a(n)
which are λn -isomorphic copies of ℓ∞ . Let τ : X → E be a norm one operator from X into a Lindenstrauss space E. Each composition τ un is a finite dimensional operator with norm at most ηn . It therefore admits, for every ε > 0, an extension a(n) Tn : ℓ∞ → E with norm at most (1 + ε)ηn . The universal property of the push-out provides a further extension to PO1 with norm (1 + ε)ηn . Choose ε ≤ ηn−1 to get a norm one extension to PO1 . Then continue this procedure inductively up to get a norm one operator on Xω . One therefore has the following theorem. Theorem 4.1. Every Banach space X can be isometrically embedded into a Lindenstrauss space L (X ) with the property that every operator τ : X → E from X into a Lindenstrauss space E can be extended to an operator T : L (X ) → E such that ∥T ∥ = ∥τ ∥. Since every separable Lindenstrauss space is 1-complemented in the Gurari˘ı space, it turns out that the Gurari˘ı space (with some suitable isometric embedding) provides a Lindenstrauss envelope for every separable Banach space. As for the C (K )-envelope, we have already remarked that the canonical isometric embedding X → C (BX ∗ ) defines a C (K )-envelope for X . Milutin’s theorem yields that for separable X the spaces C (BX ∗ ) and C [0, 1] are isomorphic. They are not isometric, however, and therefore one still cannot guarantee that C [0, 1] provides a C (K )-envelope for every separable Banach space (through some suitable isometric embedding). Nevertheless, this is true since [16] every separable C (K ) can be isometrically embedded as a 1-complemented subspace of C [0, 1]. 5. The E-envelope of X We present a general construction from which the (universally) separably injective or Lindenstrauss–Pełczyński envelope can be easily derived. Let E , X be Banach spaces. Let µ be the first ordinal whose associated cardinal has cofinality greater than densE. Set the input data of the device as:
• The ordinal µ. • For each α < µ, Xα = POα and X1 = X .
158
J.M.F. Castillo, J. Suárez / J. Math. Anal. Appl. 394 (2012) 152–158
• For each α < µ, Fα is the family of all subspaces of E. • For each α < µ, Jα denotes the family of all isometric embeddings between elements of Fα . • For each α < µ, Lα+1 is the family of all isometric embeddings of elements of Fα into Xα . Let us call the final space Xµ = Xµ (E ). This space deserves the name of E-envelope of X due to the following properties it enjoys. Proposition 5.1. Let E , X be Banach spaces. (1) For every subspace i : Y → E every operator τ : Y → Xµ (E ) can be extended to an operator T : E → Xµ with ∥T ∥ = ∥τ ∥. (2) Assume that Z has the property that for every subspace i : Y → E every operator τ : Y → Z can be extended to an operator T : E → Z with ∥T ∥ = ∥τ ∥. Then every operator τ : X → Z can be extended to an operator T : Xµ (X ) → Z with ∥T ∥ = ∥τ ∥. Proof. Assertion (1) holds by the assumption on the cofinality of µ: for every subspace Y of E and every operator τ the image τ (E ) must lie in some Xα with α < µ. Thus, every into isometry Y → Xµ is actually an into isometry Y → Xµ , being thus one of the elements of Jα which can therefore be extended to an into isometry E → POα+1 , hence to an into isometry X → Xµ through the embedding Y → E. Lemma 1.4 yields the extension of arbitrary operators. Assertion (2) is clear by transfinite induction, whose step α is as follows: the composition τ Σ Lα extends through ⊕Jα without increasing the norm. The universal property of the push out yields a simultaneous extension to Xα+1 = POα+1 . The choices E = ℓ1 , ℓ∞ , c0 and the remarks at the introduction immediately yield the following.
• The space Xω1 (ℓ1 ) is a 1-separably injective envelope of X . • The space Xω1 (ℓ∞ ) is a universally 1-separably injective envelope of X . • The space Xω1 (c0 ) is a 1-Lindenstrauss–Pełczyński envelope of X . Given λ ≥ 1, replacing Lα by the family λ−1 Lα = {λ−1 τ : τ ∈ Lα } one obtains the respective λ- envelopes. One therefore has the following theorem. Theorem 5.2. For every Banach space X and every λ ≥ 1 there exists the following.
• An isometric embedding of X into a λ-separably injective space Θ (X ) in such a way that every operator τ : X → Z from X into a λ-separably injective space Z admits an extension T : Θ (X ) → Z with ∥T ∥ = ∥τ ∥. • An isometric embedding of X into a universally λ-separably injective space U(X ) in such a way that every operator τ : X → Z from X into a universally λ-separably injective space Z admits an extension T : U(X ) → Z with ∥T ∥ = ∥τ ∥. • An isometric embedding of X into a λ-Lindenstrauss–Pełczyński space L P (X ) in such a way that every operator τ : X → Z from X into a λ-Lindenstrauss–Pełczyński space Z admits an extension T : L P (X ) → Z with ∥T ∥ = ∥τ ∥. Observe that for separable X any isometric embedding X → ℓ∞ works as a universally 1-separably injective envelope of X . Acknowledgments This research has been supported in part by project MTM2010-20190-C02-01 and the program Junta de Extremadura GR10113 IV Plan Regional I + D + i, Ayudas a Grupos de Investigación. References [1] M. Zippin, Extension of bounded linear operators, in: W.B. Johnson, J. Lindenstrauss (Eds.), Handbook of the Geometry of Banach Spaces, vol. 2, Elsevier, 2003, pp. 1703–1741. [2] J.M.F. Castillo, The hitchhiker guide to categorical Banach space theory. Part I, Extracta Math. 25 (2010) 103–149. [3] A. Avilés, F. Cabello, J.M.F. Castillo, M. González, Y. Moreno, On separably injective Banach spaces (2011), preprint arXiv:1103.6064. [4] J.M.F. Castillo, Y. Moreno, J. Suárez, On Lindenstrauss–Pelczyński spaces, Studia Math. 174 (2006) 213–231. [5] J.M.F. Castillo, Y. Moreno, J. Suárez, On the structure of Lindenstrauss–Pelczyński spaces, Studia Math. 194 (2009) 105–115. [6] A. Avilés, F. Cabello, J.M.F. Castillo, M. González, Y. Moreno, On Banach spaces of universal disposition, J. Funct. Anal. 261 (2011) 2347–2361. [7] J. Bourgain, G. Pisier, A construction of L∞ -spaces and related Banach spaces, Bol. Soc. Bras. Mat. 14 (1983) 109–123. [8] V.I. Gurari˘ı, Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces, Sibirsk. Mat. Ž 7 (1966) 1002–1013 (in Russian). [9] J.M.F. Castillo, J. Suárez, Extending operators into Lindenstrauss spaces, Israel J. Math. 169 (2009) 1–27. [10] G. Pisier, Counterexample to a conjecture of Grothendieck, Acta Math. 151 (1983) 181–208. [11] J. Suárez de la Fuente, An extension property of the Bourgain–Pisier construction. Ark. Math. (in press) http://dx.doi.org/10.1007/s11512-012-0166-8. [12] J.M.F. Castillo, M. González, Three-space problems in Banach space theory, in: Lecture Notes in Math., vol. 1667, Springer-Verlag, 1997. [13] J. Garbulinska, W. Kubis, Remarks on Gurari˘ı spaces, Extracta Math. 25 (2011). [14] W. Kubis, Fraïssé sequences — a category-theoretic approach to universal homogeneous structures (2007), arXiv:0711.1683v1. [15] J. López-Abad, S. Todorcevic, personal communication. [16] H.P. Rosenthal, The Banach spaces C (K ), in: W.B. Johnson, J. Lindenstrauss (Eds.), Handbook of the Geometry of Banach Spaces, vol. 2, Elsevier, 2003, pp. 1547–1602.