Banach spaces containing many complemented subspaces Manuel González Departamento de Matemáticas Universidad de Cantabria, Santander, Spain Workshop on Banach spaces and Banach lattices ICMAT Madrid. September 9-13, 2019 Joint work with Javier Pello (URJC, Móstoles) Manuel González (Santander) Many complemented subspaces September 13, 2019 1 / 16
Subprojective and superprojective Banach spaces NOTE: Subspaces are always closed. Definition A Banach space X is subprojective if every infinite-dim. subspace of X contains an infinite-dim. subspace complemented in X. The space X is superprojective if every infinite-codim. subspace of X is contained in an infinite-codim. subspace complemented in X. Remark X superprojective if and only if every infinite dim. quotient X / M admits an infinite dim. quotient X / N (M ⊂ N) with N complemented. Manuel González (Santander) Many complemented subspaces September 13, 2019 2 / 16
References [W-64] Robert J. Whitley. Strictly singular operators and their conjugates. Trans. Amer. Math. Soc. 113 (1964), 252–261. [OS-15] Timur Oikhberg and Eugeniu Spinu. Subprojective Banach spaces. J. Math. Anal. Appl. 424 (2015), 613–635. [GP-16] Manuel González and Javier Pello. Superprojective Banach spaces. J. Math. Anal. Appl. 437 (2016), 1140–1151. [GGP-17] Elói M. Galego, Manuel González and Javier Pello. On subprojectivity and superprojectivity of Banach spaces. Results in Math. 71 (2017), 1191–1205. [RS-18] César Ruiz and Víctor M. Sánchez. Subprojective Nakano spaces. J. Math. Anal. Appl. 458 (2018), 332–344. [GP-19] Manuel González and Javier Pello. On subprojectivity of C ( K , X ) . Proc. Amer. Math. Soc. 147 (2019), 3425–3429. [GPS-19] Manuel González, Margot Salas-Brown and Javier Pello. The perturbation classes problem on subprojective and superprojective Banach spaces. Preprint 2019. [GP-20] Manuel González and Javier Pello. Complemented subspaces of J-sums of Banach spaces. In preparation. Manuel González (Santander) Many complemented subspaces September 13, 2019 3 / 16
Basic results and examples [W64] Every subspace of a subprojective space is subprojective. 1 Every quotient of a superprojective space is superprojective. 2 Suppose X is reflexive. Then 3 ⇒ X ∗ superprojective, and X subprojective ⇐ ⇒ X ∗ subprojective. X superprojective ⇐ ℓ p (1 < p < ∞ ) is subprojective and superprojective. 1 ℓ 1 and c 0 are subprojective. 2 L p ( 0 , 1 ) subprojective ⇐ ⇒ 2 ≤ p < ∞ . 3 L p ( 0 , 1 ) superprojective ⇐ ⇒ 1 < p ≤ 2. 4 Manuel González (Santander) Many complemented subspaces September 13, 2019 4 / 16
Duality for non-reflexive spaces [GP16] X subprojective �⇒ X ∗ superprojective: X = c 0 , X ∗ = ℓ 1 . 1 ℓ 1 has a quotient ℓ 1 / M ≃ ℓ 2 . X ∗ subprojective �⇒ X superprojective: X hereditarily reflexive 2 L ∞ -space with X ∗ ≃ ℓ 1 (Bourgain Delbaen). X has a quotient X / M ≃ c 0 . NOTE: X ∗ isometric to ℓ 1 implies X superprojective. 3 Question 1 Suppose X non-reflexive. X superprojective ⇒ X ∗ subprojective? 1 X ∗ superprojective ⇒ X subprojective? 2 Manuel González (Santander) Many complemented subspaces September 13, 2019 5 / 16
Unconditional sums of Banach spaces X , Y subprojective ⇐ ⇒ X × Y subprojective [OS-15]. 1 X , Y superprojective ⇐ ⇒ X × Y superprojective [GP-16]. 2 Let E , X n ( n ∈ N ) be Banach spaces. Suppose that E admits an unconditional basis ( e n ) . We define � � ∞ � E ( X n ) := ( x n ) : x n ∈ X n for each n and � x n � e n ∈ E . n = 1 E , X n ( n ∈ N ) subprojective ⇐ ⇒ E ( X n ) subprojective [OS-15]. 1 E , X n ( n ∈ N ) superprojective ⇐ ⇒ E ( X n ) superprojective [GP-16]. 2 Manuel González (Santander) Many complemented subspaces September 13, 2019 6 / 16
Some negative criteria [GP-16] If there exists a surjective strictly singular operator Q : X → Z 1 then X is not superprojective. If there exists an strictly cosingular embedding operator J : Z → X 2 then X is not subprojective. Proposition The classes of superprojective spaces and subprojective spaces fail the three-space property. Proof. There exists an exact sequence ( Z 2 is the Kalton-Peck space) J Q 0 → ℓ 2 − → Z 2 − → ℓ 2 → 0 in which J is strictly cosingular and Q is strictly singular. Manuel González (Santander) Many complemented subspaces September 13, 2019 7 / 16
Some negative criteria [GP-16] Proposition Suppose that X contains a subspace isomorphic to ℓ 1 . Then X is not superprojective and X ∗ is not subprojective. Proof. If X contains ℓ 1 then there exists a surjective strictly singular Q : X → ℓ 2 such that Q ∗ : ℓ 2 → X ∗ is strictly cosingular. Remark This result suggests that, among the non-reflexive spaces, there are more subprojective spaces than superprojective spaces. Manuel González (Santander) Many complemented subspaces September 13, 2019 8 / 16
Tensor products Proposition ([OS-15]) X , Y ∈ { c 0 , ℓ p 1 ≤ p < ∞} ⇒ X ˆ ⊗ π Y and X ˆ ⊗ ǫ Y subprojective. 1 2 ≤ p , q < ∞ ⇒ L p ( 0 , 1 )ˆ ⊗ ǫ L q ( 0 , 1 ) subprojective. 2 Proposition ([GP-16]) X , Y ∈ { c 0 , ℓ p 1 < p < ∞} ⇒ X ˆ ⊗ ǫ Y superprojective. 1 Let 1 < p , q < ∞ . Then 2 ℓ p ˆ ⊗ π ℓ q superprojective ⇔ p > q / ( q − 1 ) ⇔ ℓ p ˆ ⊗ π ℓ q reflexive. For 1 < p , q ≤ 2 , L p ( 0 , 1 )ˆ ⊗ π L q ( 0 , 1 ) is not superprojective. 3 Question 2 Is L p ( 0 , 1 )ˆ ⊗ π L q ( 0 , 1 ) subprojective when 2 ≤ p , q < ∞ ? 1 Is L p ( 0 , 1 )ˆ ⊗ ǫ L q ( 0 , 1 ) superprojective when 1 < p , q ≤ 2? 2 Manuel González (Santander) Many complemented subspaces September 13, 2019 9 / 16
Spaces C ( K , X ) and L p ( X ) Let K be a compact space. C ( K ) subprojective ⇐ ⇒ C ( K ) superprojective ⇐ ⇒ K scattered. Theorem (GP-19) ⇒ C ( K , X ) ≡ C ( K )ˆ C ( K ) and X subprojective = ⊗ ǫ X subprojective. Question 3 C ( K ) and X superprojective ⇒ C ( K , X ) superprojective? Proposition (GP-16) X superprojective = ⇒ C ([ 0 , λ ] , X ) superprojective. Observation. (F .L. Hernández) [Y. Raynaud 1985]: For 2 < p < q < ∞ , L p ( L q ) is not subprojective (while L p and L q are). For 1 < s < r < 2, L r ( L s ) is not superprojective (while L r and L s are). Manuel González (Santander) Many complemented subspaces September 13, 2019 10 / 16
Properties implying superprojectivity [GGP-17] Definition We say that X satisfies P weak if each non-weakly compact operator T : X → Y is an isomorphism on a copy of c 0 and X ∗ is hereditarily ℓ 1 . We say that X satisfies P strong if each non-compact operator T : X → Y is an isomorphism on a copy of c 0 . Proposition X satisfies P strong ⇒ X satisfies P weak ⇒ X is superprojective. 1 If X n (n ∈ N ) satisfy P strong then c 0 ( X n ) satisfies P strong . 2 If X and Y satisfy P strong then X ˆ ⊗ π Y satisfies P strong . 3 Manuel González (Santander) Many complemented subspaces September 13, 2019 11 / 16
Properties implying superprojectivity [GGP-17] Examples Isometric preduals of ℓ 1 (Γ) satisfy P strong . 1 C ( K ) spaces with K scattered satisfy P strong . 2 The Hagler space JH satisfies P strong . 3 The Schreier space S and the predual of the Lorentz space 4 d ( w , 1 ) satisfy P weak but not P strong . In fact S ˆ ⊗ π S is not superprojective. Question 4 Find new examples of sub(super)projective Banach spaces. Manuel González (Santander) Many complemented subspaces September 13, 2019 12 / 16
J -sums of Banach spaces I [GP-20] dim J ∗∗ / J = 1. J : James’ space. J and J ∗ are subprojective. By duality, J and J ∗ are superprojective. [S.F. Bellenot. The J -sum of Banach spaces. J. Funct. Analysis (1982)] i 1 i 2 i 3 ( X 1 , � · � 1 ) − → ( X 2 , � · � 2 ) − → ( X 3 , � · � 3 ) − → · · · � i k � ≤ 1 . J ( X n ) lim = { ( x i ) i ∈ N : x i ∈ X i , � ( x i ) � J : < ∞} . J = sup { � k − 1 i = 1 � x p ( i + 1 ) − φ p ( i + 1 ) where � ( x i ) � 2 ( x p ( i ) ) � 2 p ( i + 1 ) } , p ( i ) and the sup is taken over k ∈ N and p ( 1 ) < · · · < p ( k ) . Moreover, J ( X n ) = { ( x i ) ∈ J ( X n ) lim : lim i →∞ � ( x i ) � i : < ∞} . Manuel González (Santander) Many complemented subspaces September 13, 2019 13 / 16
J -sums of Banach spaces II Observation [Bellenot]. J ( X n ) lim is a Banach space and J ( X n ) is a subspace of J ( X n ) lim . 1 If each X n is reflexive, then J ( X n ) ∗∗ ≡ J ( X n ) lim . 2 Theorem If each X n is subprojective , then so is J ( X n ) . 1 If each X n is superprojective, then so is J ( X n ) . 2 If each X ∗ n is subprojective, then so is J ( X n ) ∗ . 3 Manuel González (Santander) Many complemented subspaces September 13, 2019 14 / 16
J -sums of Banach spaces III Special case : ( X n ) is an increasing sequence of subspaces of a Banach space Y with ∪ ∞ n = 1 X n dense in Y , and i k : X k → X k + 1 is the inclusion. Observation [Bellenot]. The expression U ( x k ) = lim k →∞ x k defines a surjective operator 1 U : J ( X n ) lim → Y with kernel J ( X n ) . If each X n is reflexive, then J ( X n ) ∗∗ / J ( X n ) ≡ J ( X n ) lim / J ( X n ) ≡ Y . 2 Theorem If Y is subprojective, then J ( X n ) is subprojective. Manuel González (Santander) Many complemented subspaces September 13, 2019 15 / 16
Thank you for your attention. Manuel González (Santander) Many complemented subspaces September 13, 2019 16 / 16
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