Uniform classification of classical Banach spaces B unyamin Sar - PowerPoint PPT Presentation

Uniform classification of classical Banach spaces B¨ unyamin Sarı University of North Texas BWB 2014

Uniform classification question A bijection φ : X → Y is a uniform homeomorphism if both φ and φ − 1 are uniformly continuous.

Uniform classification question A bijection φ : X → Y is a uniform homeomorphism if both φ and φ − 1 are uniformly continuous. Basic questions: Suppose X is uniformly homeomorphic to Y . Are they linearly isomorphic? If not, how much of the linear structure is preserved?

Uniform classification question A bijection φ : X → Y is a uniform homeomorphism if both φ and φ − 1 are uniformly continuous. Basic questions: Suppose X is uniformly homeomorphic to Y . Are they linearly isomorphic? If not, how much of the linear structure is preserved? Ribe ’76. The local structure is preserved: There exists K = K ( φ ) such that every finite dimensional subspace of X K -embeds into Y , and vice versa.

Classical spaces Johnson-Lindenstrauss-Schechtman ’96 Suppose X is uniformly homeomorphic to ℓ p for 1 < p < ∞ . Then X is isomorphic to ℓ p .

Classical spaces Johnson-Lindenstrauss-Schechtman ’96 Suppose X is uniformly homeomorphic to ℓ p for 1 < p < ∞ . Then X is isomorphic to ℓ p . Godefroy-Kalton-Lancien ’00 If X is Lipschitz isomorphic c 0 , then X is isomorphic to c 0 . If X is uniformly homeomorphic to c 0 , then X is ‘almost’ isomorphic to c 0 .

Classical spaces Johnson-Lindenstrauss-Schechtman ’96 Suppose X is uniformly homeomorphic to ℓ p for 1 < p < ∞ . Then X is isomorphic to ℓ p . Godefroy-Kalton-Lancien ’00 If X is Lipschitz isomorphic c 0 , then X is isomorphic to c 0 . If X is uniformly homeomorphic to c 0 , then X is ‘almost’ isomorphic to c 0 . Open for ℓ 1 (Lipschitz case too)

Idea of the proof for 1 < p < ∞ case Enough to show ℓ 2 � ֒ → X (follows from Ribe and Johnson-Odell)

Idea of the proof for 1 < p < ∞ case Enough to show ℓ 2 � ֒ → X (follows from Ribe and Johnson-Odell) For 1 ≤ p < 2 Midpoint technique Enflo ‘69, Bourgain ‘87

Idea of the proof for 1 < p < ∞ case Enough to show ℓ 2 � ֒ → X (follows from Ribe and Johnson-Odell) For 1 ≤ p < 2 Midpoint technique Enflo ‘69, Bourgain ‘87 For 2 < p < ∞ Gorelik principle Gorelik ‘94

Idea of the proof for 1 < p < ∞ case Enough to show ℓ 2 � ֒ → X (follows from Ribe and Johnson-Odell) For 1 ≤ p < 2 Midpoint technique Enflo ‘69, Bourgain ‘87 For 2 < p < ∞ Gorelik principle Gorelik ‘94 Alternatively, for 2 < p < ∞ Asymptotic smoothness Kalton-Randrianarivony ‘08

Idea of the proof for 1 < p < ∞ case Enough to show ℓ 2 � ֒ → X (follows from Ribe and Johnson-Odell) For 1 ≤ p < 2 Midpoint technique Enflo ‘69, Bourgain ‘87 For 2 < p < ∞ Gorelik principle Gorelik ‘94 Alternatively, for 2 < p < ∞ Asymptotic smoothness Kalton-Randrianarivony ‘08 We will give another.

Result Theorem. Suppose φ : X → Y is a uniform homeomorphism and Y is reflexive. Then there exists K = K ( φ ) such that for all n and all asymptotic spaces ( x i ) n i =1 of X and all scalars ( a i ) n i =1 , we have n n � � � a i x i � ≤ K sup � a i y i � i =1 i =1 where sup is over all ( y i ) n i =1 asymptotic spaces of Y .

Result Theorem. Suppose φ : X → Y is a uniform homeomorphism and Y is reflexive. Then there exists K = K ( φ ) such that for all n and all asymptotic spaces ( x i ) n i =1 of X and all scalars ( a i ) n i =1 , we have n n � � � a i x i � ≤ K sup � a i y i � i =1 i =1 where sup is over all ( y i ) n i =1 asymptotic spaces of Y . If Y = ℓ p , then this means n n � � | a i | p ) 1 /p . � a i x i � ≤ K ( i =1 i =1 Thus, X cannot contain ℓ 2 if p > 2.

Asymptotic structure Maurey-Milman-Tomczak-Jaegermann ’94 Let X be a Banach space with a normalized basis (or a minimal system) ( u i ). Write n < x < y if n < min supp x < max supp x < min supp y .

Asymptotic structure Maurey-Milman-Tomczak-Jaegermann ’94 Let X be a Banach space with a normalized basis (or a minimal system) ( u i ). Write n < x < y if n < min supp x < max supp x < min supp y . An n -dimensional space with basis ( e i ) n 1 is called an asymptotic space of X , write ( e i ) n 1 ∈ { X } n , if for all ε > 0 ∀ m 1 ∃ m 1 < x 1 ∀ m 2 ∃ m 2 < x 2 . . . ∀ m n ∃ m n < x n such that the resulting blocks (called permissible ) satisfy 1+ ε ( x i ) n ∼ ( e i ) n 1 . 1

Asymptotic tree ( e i ) n 1 ∈ { X } n means that for all ε > 0 there exists a block tree of n -levels T n = { x ( k 1 , k 2 , . . . , k j ) : 1 ≤ j ≤ n } so that every branch ( x ( k 1 ) , x ( k 1 , k 2 ) , . . . , x ( k 1 , . . . , k n )) is (1 + ε )-equivalent to ( e i ) n 1 .

Asymptotic- ℓ p spaces X is asymptotic - ℓ p ( asymptotic - c 0 for p = ∞ ), if there exists K K ≥ 1 such that for all n and ( e i ) n 1 ∈ { X } n , ( e i ) n ∼ uvb ℓ n p . 1

Asymptotic- ℓ p spaces X is asymptotic - ℓ p ( asymptotic - c 0 for p = ∞ ), if there exists K K ≥ 1 such that for all n and ( e i ) n 1 ∈ { X } n , ( e i ) n ∼ uvb ℓ n p . 1 ℓ p is asymptotic- ℓ p .

Asymptotic- ℓ p spaces X is asymptotic - ℓ p ( asymptotic - c 0 for p = ∞ ), if there exists K K ≥ 1 such that for all n and ( e i ) n 1 ∈ { X } n , ( e i ) n ∼ uvb ℓ n p . 1 ℓ p is asymptotic- ℓ p . L p is not. Indeed, every C-unconditional ( x i ) n 1 ⊂ L p is CK p -equivalent to some asymptotic space of L p .

Asymptotic- ℓ p spaces X is asymptotic - ℓ p ( asymptotic - c 0 for p = ∞ ), if there exists K K ≥ 1 such that for all n and ( e i ) n 1 ∈ { X } n , ( e i ) n ∼ uvb ℓ n p . 1 ℓ p is asymptotic- ℓ p . L p is not. Indeed, every C-unconditional ( x i ) n 1 ⊂ L p is CK p -equivalent to some asymptotic space of L p . Tsirelson space T is asymptotic- ℓ 1 .

Asymptotic- ℓ p spaces X is asymptotic - ℓ p ( asymptotic - c 0 for p = ∞ ), if there exists K K ≥ 1 such that for all n and ( e i ) n 1 ∈ { X } n , ( e i ) n ∼ uvb ℓ n p . 1 ℓ p is asymptotic- ℓ p . L p is not. Indeed, every C-unconditional ( x i ) n 1 ⊂ L p is CK p -equivalent to some asymptotic space of L p . Tsirelson space T is asymptotic- ℓ 1 . T ∗ is asymptotic- c 0 .

Envelope functions Define the upper envelope function r X on c 00 by n � r X ( a 1 , . . . , a n ) = sup � a i e i � ( e i ) n 1 ∈{ X } n i and the lower envelope g X by n � g X ( a 1 , . . . , a n ) = inf � a i e i � ( e i ) n 1 ∈{ X } n i

Envelope functions Define the upper envelope function r X on c 00 by n � r X ( a 1 , . . . , a n ) = sup � a i e i � ( e i ) n 1 ∈{ X } n i and the lower envelope g X by n � g X ( a 1 , . . . , a n ) = inf � a i e i � ( e i ) n 1 ∈{ X } n i X is asymptotic- ℓ p iff g X ≃ � . � p ≃ r X .

Envelope functions Define the upper envelope function r X on c 00 by n � r X ( a 1 , . . . , a n ) = sup � a i e i � ( e i ) n 1 ∈{ X } n i and the lower envelope g X by n � g X ( a 1 , . . . , a n ) = inf � a i e i � ( e i ) n 1 ∈{ X } n i X is asymptotic- ℓ p iff g X ≃ � . � p ≃ r X . r X ≃ � . � ∞ implies X is asymptotic- c 0 .

The upper envelope is invariant Theorem. Suppose φ : X → Y is uniform homeomorphism, and X and Y are reflexive. Then there exists K = K ( φ ) such that for all scalars a = ( a i ) ∈ c 00 , we have 1 K r Y ( a ) ≤ r X ( a ) ≤ Kr Y ( a ) .

The upper envelope is invariant Theorem. Suppose φ : X → Y is uniform homeomorphism, and X and Y are reflexive. Then there exists K = K ( φ ) such that for all scalars a = ( a i ) ∈ c 00 , we have 1 K r Y ( a ) ≤ r X ( a ) ≤ Kr Y ( a ) . Corollary. Suppose X is uniformly homeomorphic to a reflexive asymptotic- c 0 space. Then X is asymptotic- c 0 .

The upper envelope is invariant Theorem. Suppose φ : X → Y is uniform homeomorphism, and X and Y are reflexive. Then there exists K = K ( φ ) such that for all scalars a = ( a i ) ∈ c 00 , we have 1 K r Y ( a ) ≤ r X ( a ) ≤ Kr Y ( a ) . Corollary. Suppose X is uniformly homeomorphic to a reflexive asymptotic- c 0 space. Then X is asymptotic- c 0 . Example. T ∗

The main technical theorem Theorem. Suppose φ : X → Y is a uniform homeomorphism and Y is reflexive. Then for all ( e i ) k 1 ∈ { X } k , integers ( a i ) k 1 and 1+ ε ε > 0, there exist permissible ( x i ) k 1 in X with ( x i ) k ∼ ( e i ) k 1 and 1 permissible tuple ( h i / � h i � ) k 1 in Y with � h i � ≤ K | a i | ( K depends only on φ ) such that � k k � � � � � � � � φ a i x i − h i � ≤ ε. � � � � i =1 i =1