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Principle of local reflexivity, respecting subspaces, and approximation properties of pairs

Eve Oja

University of Tartu, Tartu, Estonia Estonian Academy of Sciences

X – Banach space, λ ≥ 1 Recall: X has λ-BAP: ∀E ⊂ X, dim E < ∞, ∀ε > 0 ∃S ∈ F(X) := F(X, X) such that S ≤ λ + ε and Sx = x, x ∈ E. Equivalent to ([JRZ]=Johnson, Rosenthal, Zippin; Israel J. Math, 1971): ∃(Sα) ⊂ F(X) such that lim sup Sα ≤ λ and Sα → IX pointwise. Y ⊂ X, Y – closed subspace [FJP]=Figiel, Johnson, Pełczy´ nski, Some approximation properties of Banach spaces and Banach lattices, Israel J. Math, 2011: The pair (X, Y) has λ-BAP: ∀E ⊂ X, dim E < ∞, ∀ε > 0 ∃S ∈ F(X) such that S(Y) ⊂ Y, S ≤ λ + ε, and Sx = x, x ∈ E. X – λ-BAP ⇔ (X, X) – λ-BAP ⇔ (X, {0}) – λ-BAP

Y ⊂ X ⇒ Y ⊥ := {x∗ ∈ X ∗ : x∗(y) = 0 ∀y ∈ Y} ⊂ X ∗ S(Y) ⊂ Y ⇔ S∗(Y ⊥) ⊂ Y ⊥ For reflexive X: (X, Y) – BAP⇔ (X ∗, Y ⊥) – BAP In general: (X, Y) – BAP⇒ (X ∗, Y ⊥) – BAP Enflo, James, Lindenstrauss: X – BAP⇒ X ∗ – BAP Grothendieck (essentially): X – λ-BAP⇐ X ∗ – λ-BAP easy with the PLR from [Johnson;TAMS,1971] or [JRZ]) [Oja, Treialt; Stud.Math,2013]: (X, Y) – λ-BAP⇐ (X ∗, Y ⊥) – λ-BAP the PLR seems not working

[Oja, Treialt, Stud. Math, 2013] :

• (X, Y) – λ-BAP ⇐ (X ∗, Y ⊥) – λ-BAP

the PLR seems not working Proof relies on [Oja, JMAA, 2006]; it does not work if the BAP is given by projections.

• Find a “working” PLR (respecting subspaces) which would give an

easy proof and would apply to the case of projections! BAP given by projections =: π-property X has πλ-property: ∀E ⊂ X, dim E < ∞, ∀ε > 0 ∃ projection P ∈ F(X) such that P ≤ λ + ε and Px = x, x ∈ E. 50th anniversary: Lindenstrauss’s Memoir, 1964. Important contributions: Michael & Pełczy´ nski, Israel J. Math, 1966 (π1-property (π∞

1 -property: ranP ∼

= ℓn

∞)); Johnson, Illinois J. Math, 1970; [JRZ];

Pełczy´ nski & Rosenthal, Studia Math, 1975; etc.

Th.1 [JRZ]: (a) X – π-property ⇐ X ∗ – π-property (b) X – π-property ⇒ X ∗ – π-property X ∗ – BAP Proof relies on a strong form of PLR involving projections.

• Extend JRZ Theorem 1 to (X, Y)!
• Find a PLR respecting subspaces and involving projections!

(X, Y) has πλ-property: ∀E ⊂ X, dim E < ∞, and ∀ε > 0 ∃ projection P ∈ F(X) such that P(Y) ⊂ Y, P ≤ λ + ε, and Px = x, x ∈ E. (X, Y) has πλ-duality property: ∀E ⊂ X, dim E < ∞, and ∀F ⊂ X ∗, dim F < ∞, and ∀ε > 0 ∃ projection P ∈ F(X) such that P(Y) ⊂ Y, P ≤ λ + ε, and Px = x, x ∈ E, and P∗x∗ = x∗, x∗ ∈ F. X – πλ-(dual)prop⇔(X, X) – πλ-(dual)prop⇔(X, {0}) – πλ-(dual)prop

PLR-1 respecting subspaces: X, Z – Banach spaces; U ⊂ X, V ⊂ Z – closed subspaces; let S ∈ F(Z ∗, X ∗) satisfy S(V ⊥) ⊂ U⊥. If F ⊂ Z ∗, dim F < ∞, and ε > 0, then ∃T ∈ F(X, Z) satisfying T(U) ⊂ V such that 1◦

• T − S
• < ε,

2◦ T ∗z∗ = Sz∗, z∗ ∈ F, 2◦◦ ranT ∗ = ranS, 3◦ T ∗∗x∗∗ = S∗x∗∗ whenever S∗x∗∗ ∈ Z. When X = Z and S is a projection, also T is a projection. Proof relies on Grothendieck’s description (X ⊗ Z)∗ = I(X, Z ∗), integral operators, equipped with their integral norms. (X ⊗ Z ⊂ (F(X ∗, Z), · ); (x ⊗ z)(x∗)=x∗(x)z, x∗ ∈ X ∗.) (Via duality A, x ⊗ z = (Ax)(z), A ∈ I(X, Z ∗).)

Immediate applications of PLR-1 respecting subspaces:

• Cor. 1: λ-BAP (πλ-property) of (X ∗, Y ⊥) is given by conjugate
• perators (projections).

Standard arguments (incl. passing to convex combinations of approximating operators) give a refinement for BAP (but not for π-property):

• Cor. 2 [Oja–Treialt]: (X ∗, Y ⊥) – λ-BAP⇒ (X, Y) – λ-BdualityAP:

∀E ⊂ X, dim E < ∞, and ∀F ⊂ X ∗, dim F < ∞, and ∀ε > 0 ∃S ∈ F(X) such that S(Y) ⊂ Y, S ≤ λ + ε, and Sx = x, x ∈ E, and S∗x∗ = x∗, x∗ ∈ F. >From Cor. 1 and Cor. 2, we get the following Theorem:

Th.: (a) (X ∗, Y ⊥) – πλ-prop. ⇒ (X, Y) – πλµ+λ+µ-dual. prop. (X, Y) – µ-BAP (b) (X, Y) – πλ-prop.

• ⇒ (X, Y) – πλµ+λ+µ-dual. prop.

(X ∗, Y ⊥) – µ-BAP Proof: (b) Let E ⊂ X, dim E < ∞, F ⊂ X ∗, dim F < ∞, and let ε > 0. Choose δ > 0 such that (2 + λ + µ)δ + δ2 < ε. Cor. 2: (X, Y) – µ-BdualityAP: ∃S ∈ F(X) such that S(Y) ⊂ Y, S ≤ µ + δ, Sx = x, x ∈ E, and S∗x∗ = x∗, x∗ ∈ F. (X, Y) – πλ-prop.: ∃P ∈ F(X) such that P(Y) ⊂ Y, P ≤ λ + δ, and Px = x, x ∈ ranS. Then PS = S. Hence, Q := P + S − SP ∈ F(X) is a needed projection. (a) Similar; uses Cor. 1: πλ-prop. given by conj. op.

Th.: (a) (X ∗, Y ⊥) – πλ-prop.

• ⇒ (X, Y) – πλµ+λ+µ-dual. prop.

(X, Y) – µ-BAP (b) (X, Y) – πλ-prop.

• ⇒ (X, Y) – πλµ+λ+µ-dual. prop.

(X ∗, Y ⊥) – µ-BAP [Lissitsin, Oja; JMAA, 2011]: If X ∗ or X ∗∗ has RNP , then (X ∗, Y ⊥) – AP⇒ (X ∗, Y ⊥) – 1-BAP . Cor.: If X ∗ or X ∗∗ has RNP , then (X ∗, Y ⊥) – πλ-prop. ⇒ (X, Y) – π2λ+1-duality prop.

• Th. (a), applied 2×: (X ∗∗, Y ⊥⊥) – πλ-prop. ⇒ (X, Y) – πµ-prop.

with µ = (λ2 + 2λ)2 + 2(λ2 + 2λ)

• Th. 2 [JRZ]: X ∗∗ – πλ-prop. ⇒ X – πλ-prop.

We can extend this using PLR-2 respecting subspaces (weak version): X, Z – Banach spaces; U ⊂ X, V ⊂ Z – closed subspaces. Let S ∈ F(X ∗∗, Z ∗∗) satisfy S(U⊥⊥) ⊂ V ⊥⊥. If E ⊂ X ∗∗, dim E < ∞, and F ⊂ Z ∗, dim F < ∞, and ε > 0, then ∃T ∈ F(X, Z) satisfying T(U) ⊂ V such that 1◦

• T − S
• < ε,

2◦ x∗∗(T ∗z∗) = (Sx∗∗)(z∗), x∗∗ ∈ E and z∗ ∈ F, 3◦ T ∗∗x∗∗ = Sx∗∗ whenever Sx∗∗ ∈ Z. When X = Z and S is a projection, also T is a projection. Proof is immediate: apply 2× the PLR-1 resp. subsp. Cor.: (X ∗∗, Y ⊥⊥) – πλ-prop ⇒ (X, Y) – πλ-prop.

“Moreover” part to the weak version (by “enlarging” argument): PLR-2 respecting subspaces: X, Z – Banach spaces; U ⊂ X, V ⊂ Z – closed subspaces. Let S ∈ F(X ∗∗, Z ∗∗) satisfy S(U⊥⊥) ⊂ V ⊥⊥. If E ⊂ X ∗∗, dim E < ∞, and F ⊂ Z ∗, dim F < ∞, and ε > 0, then ∃T ∈ F(X, Z) satisfying T(U) ⊂ V such that 1◦

• T − S
• < ε,

2◦ x∗∗(T ∗z∗) = (Sx∗∗)(z∗), x∗∗ ∈ E and z∗ ∈ F, 3◦ T ∗∗x∗∗ = Sx∗∗ whenever Sx∗∗ ∈ Z. When X = Z and S is a projection, also T is a projection. Moreover, if S|E is one-to-one, then also T ∗∗|E is, and 1◦◦ (T ∗∗|E)−1 < (S|E)−1 + ε. PLR in [JRZ]: X = E ⊂ Z ∗∗, S = Id : E → Z ∗∗, U = {0}, V = {0}. Bellenot’s PLR [J. Funct. Anal, 1984]: PLR + T(E ∩ V ⊥⊥) ⊂ V. Proof: X, S as above, U = E ∩ V ⊥⊥ (⇒ S(U⊥⊥) = U ⊂ V ⊥⊥).

PLR-1 respecting subspaces easily follows from: Lemma: G ⊂ X ∗, V ⊂ Z; let T ∈ X ⊗ Z ∗∗ satisfy T(G) ⊂ V ⊥⊥. If F ⊂ Z ∗, dim F < ∞, then ∃(Tα) ⊂ X ⊗ Z satisfying Tα(G) ⊂ V, for all α, such that 1◦ Tα → T, 2◦ T ∗

αz∗ → T ∗z∗, z∗ ∈ Z ∗; and T ∗ αz∗ = T ∗z∗, z∗ ∈ F, for all α,

3◦ Tαx∗ = Tx∗ for all α whenever Tx∗ ∈ Z. Recall: X ⊗ Z ⊂ F(X ∗, Z); (x ⊗ z)(x∗) = x∗(x)z, x∗ ∈ X ∗. Proof of Lemma: (1) R := {R ∈ X ⊗ Z : R(G) ⊂ V}, S := {S ∈ X ⊗ Z ∗∗ : S(G) ⊂ V ⊥⊥} R⊥ ⊂ (X ⊗ Z)∗ = I(X, Z ∗)

J

→ I(X, Z ∗∗∗) = (X ⊗ Z ∗∗)∗ ⊃ S⊥ J(A)=jZ ∗A, A ∈ I(X, Z ∗), (jZ ∗ : Z ∗ → Z ∗∗∗ is can. emb.) J – isometry into; J(R⊥) ⊂ S⊥ Φ(A + R⊥)=J(A) + S⊥, A ∈ I(X, Z ∗) R∗ = I(X, Z ∗)/R⊥ Φ → I(X, Z ∗∗∗)/S⊥ = S∗ Φ∗(T) ∈ TBR∗∗ ⇒ ∃(Tα) ⊂ R, i.e., Tα(G) ⊂ V for all α, with 1◦, T ∗

αz∗ → T ∗z∗, z∗ ∈ Z ∗, and Tαx∗ → Tx∗ whenever Tx∗ ∈ Z.

(2) Make convergences “constant” (where needed) using a perturbation argument from [Oja, Põldvere; PAMS, 2007], inspired by [JRZ]. For the details, please see:

• E. Oja, Principle of local reflexivity respecting subspaces, Adv. Math.

258 (2014) 1–12.