Principle of local reflexivity, respecting subspaces, and - PowerPoint PPT Presentation

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): ∃ ( 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, 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 ⊥ ( 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 from [Johnson;TAMS,1971] or [JRZ]) ( 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 . 50th anniversary: Lindenstrauss’s Memoir, 1964. Important contributions: Michael & Pełczy´ nski, Israel J. Math, 1966 ( π 1 -property 1 -property: ran P ∼ ( π ∞ = ℓ n ∞ )); Johnson, Illinois J. Math, 1970; [JRZ]; Pełczy´ nski & Rosenthal, Studia Math, 1975; etc.

⇐ 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 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 ( 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.

(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 [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 ⊥ ) – π λ -prop. ⇒ ( X , Y ) – π 2 λ + 1 -duality prop.

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 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 ⊥⊥ ) – π λ -prop ⇒ ( X , Y ) – π λ -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 � < ε , 1 ◦ � � � � T � − � S � x ∗∗ ( T ∗ z ∗ ) = ( Sx ∗∗ )( z ∗ ) , x ∗∗ ∈ E and z ∗ ∈ F , 2 ◦ T ∗∗ x ∗∗ = Sx ∗∗ whenever Sx ∗∗ ∈ Z . 3 ◦ 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 � ( 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: E. Oja, Principle of local reflexivity respecting subspaces, Adv. Math. 258 (2014) 1–12.