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 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: • Motivations for some more forms of PLR • These forms and their applications

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: ∃ ( S α ) ⊂ F ( X ) such that lim sup � S α � ≤ λ and S α → I X 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 ⊥ ( X , Y ) – BAP ⇔ ( X ∗ , Y ⊥ ) – BAP For reflexive X : ( X , Y ) – BAP �⇒ ( X ∗ , Y ⊥ ) – BAP In general : X – BAP �⇒ X ∗ – BAP Enflo, James, Lindenstrauss : X – λ -BAP ⇐ X ∗ – λ -BAP Grothendieck (essentially) : easy with the PLR ( X , Y ) – λ -BAP ⇐ ( X ∗ , Y ⊥ ) – λ -BAP [ Oja, Treialt; Stud.Math,2013 ] : 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 π λ -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:

⇐ X ∗ – π -property Th.1 [JRZ]: (a) 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 π λ -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 π λ -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 – π λ -(dual)prop ⇔ ( X , X ) – π λ -(dual)prop ⇔ ( X , { 0 } ) – π λ -(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 � < ε , 1 ◦ � � � � T � − � S � T ∗ z ∗ = Sz ∗ , z ∗ ∈ F , 2 ◦ ran T ∗ = ran S , 2 ◦◦ T ∗∗ x ∗∗ = S ∗ x ∗∗ whenever S ∗ x ∗∗ ∈ Z . 3 ◦ 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: Cor. 1: λ -BAP ( π λ -property) of ( X ∗ , Y ⊥ ) is given by conjugate operators (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:

(a) ( X ∗ , Y ⊥ ) – π λ -prop. � Th.: ⇒ ( X , Y ) – π λµ + λ + µ -dual. prop. ( X , Y ) – µ -BAP � (b) ( X , Y ) – π λ -prop. ⇒ ( X , Y ) – π λµ + 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 + δ 2 < 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 ) – π λ -prop.: ∃ P ∈ F ( X ) such that P ( Y ) ⊂ Y , � P � ≤ λ + δ , and Px = x , x ∈ ran S . Then PS = S . Hence, Q := P + S − SP ∈ F ( X ) is a needed projection. (a) Similar; uses Cor. 1: π λ -prop. given by conj. op.

Th. (a), applied 2 × : ( X ∗∗ , Y ⊥⊥ ) – π λ -prop. ⇒ ( X , Y ) – π µ -prop. with µ = ( λ 2 + 2 λ ) 2 + 2 ( λ 2 + 2 λ ) Th. 2 [JRZ]: X ∗∗ – π λ -prop. ⇒ X – π λ -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 � < ε , � � 1 ◦ � � T � − � S � x ∗∗ ( T ∗ z ∗ ) = ( Sx ∗∗ )( z ∗ ) , x ∗∗ ∈ E and z ∗ ∈ F , 2 ◦ T ∗∗ x ∗∗ = Sx ∗∗ whenever Sx ∗∗ ∈ Z . 3 ◦ 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 ⊥⊥ ) – π λ -prop ⇒ ( X , Y ) – π λ -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 � < ε , 1 ◦ � � � � T � − � S � x ∗∗ ( T ∗ z ∗ ) = ( Sx ∗∗ )( z ∗ ) , x ∗∗ ∈ E and z ∗ ∈ F , 2 ◦ T ∗∗ x ∗∗ = Sx ∗∗ whenever Sx ∗∗ ∈ Z . 3 ◦ 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 � ( T ∗∗ | E ) − 1 � < � ( S | E ) − 1 � + ε . 1 ◦◦ 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 ∃ ( T α ) ⊂ X ⊗ Z satisfying T α ( G ) ⊂ V , for all α , such that 1 ◦ � T α � → � T � , α z ∗ → T ∗ z ∗ , z ∗ ∈ Z ∗ ; and T ∗ α z ∗ = T ∗ z ∗ , z ∗ ∈ F , for all α , 2 ◦ T ∗ T α x ∗ = Tx ∗ for all α whenever Tx ∗ ∈ Z . 3 ◦ 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 ∗ ) → I ( X , Z ∗∗∗ ) = ( X ⊗ Z ∗∗ ) ∗ ⊃ S ⊥ J J ( A )= j Z ∗ A , A ∈ I ( X , Z ∗ ) , ( j Z ∗ : 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 ) ∈ � T � B R ∗∗ ⇒ ∃ ( T α ) ⊂ R , i.e., T α ( G ) ⊂ V for all α , with 1 ◦ , α z ∗ → T ∗ z ∗ , z ∗ ∈ Z ∗ , and T α x ∗ → Tx ∗ whenever Tx ∗ ∈ Z . T ∗

(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.