Principle of local reflexivity respecting subspaces, and - - PowerPoint PPT Presentation
Principle of local reflexivity respecting subspaces, and approximation properties given by projections Eve Oja Rafael Pays birthday conference, 2015 PLR was discovered by Lindenstrauss and Rosenthal in 1969. Improved in [JRZ]=Johnson,
Principle of local reflexivity respecting subspaces, and approximation properties given by projections Eve Oja Rafael Payá’s birthday conference, 2015
PLR was discovered by Lindenstrauss and Rosenthal in 1969. Improved in [JRZ]=Johnson, Rosenthal, Zippin; Israel J. Math, 1971. · · · For the most advanced forms of PLR and thorough analysis, see: Behrends, Studia Math. 100 (1991) 109–128. PLAN of the talk:
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: ∃(Sα) ⊂ F(X) such that lim sup Sα ≤ λ and Sα → IX pointwise.) Y ⊂ X, Y – closed subspace [FJP]=Figiel, Johnson, Pełczy´ nski; 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 [Oja, Treialt; Stud.Math,2013]: (X, Y) – λ-BAP⇐ (X ∗, Y ⊥) – λ-BAP the PLR seems not working
[Oja, Treialt, Stud. Math, 2013] :
the PLR seems not working Proof relies on [Oja, JMAA, 2006]; it does not work if the BAP is given by projections.
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. [JRZ] contains the following lifting theorem for the π-property:
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.
(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◦
2◦ T ∗z∗ = Sz∗, z∗ ∈ F, 2◦◦ ranT ∗ = ranS, 3◦ T ∗∗x∗∗ = S∗x∗∗ whenever S∗x∗∗ ∈ Z. When Z = X 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:
Standard arguments (incl. passing to convex combinations of approximating operators) give a refinement for BAP (but not for π-property):
∀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 ⊥) – 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.
with µ = (λ2 + 2λ)2 + 2(λ2 + 2λ)
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◦
2◦ x∗∗(T ∗z∗) = (Sx∗∗)(z∗), x∗∗ ∈ E and z∗ ∈ F, 3◦ T ∗∗x∗∗ = Sx∗∗ whenever Sx∗∗ ∈ Z. When Z = X 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 for 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◦
2◦ x∗∗(T ∗z∗) = (Sx∗∗)(z∗), x∗∗ ∈ E and z∗ ∈ F, 3◦ T ∗∗x∗∗ = Sx∗∗ whenever Sx∗∗ ∈ Z. When Z = X 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: Oja, Principle of local reflexivity respecting subspaces, Adv. Math. 258 (2014) 1–12.