How does the distortion of linear embedding of C 0 ( K ) into C 0 ( , - PowerPoint PPT Presentation

Bibliography How does the distortion of linear embedding of C 0 ( K ) into C 0 ( Γ, X ) spaces depend on the height of K? Leandro Candido Batista Joint work with El´ oi Medina Galego University of S˜ ao Paulo, Department of Mathematics, IME lc@ime.usp.br Brazilian Conference on General Topology and Set Theory, S˜ ao Sebasti˜ ao, August 12-16, 2013 Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces L. Candido, E. M. Galego How does the distortion of linear embedding of C 0 ( K ) into C 0 ( Γ, X ) spaces depend on the height of K? , J. Math. Anal. Appl. 402 (2013), 185–190. Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces L. Candido, E. M. Galego How does the distortion of linear embedding of C 0 ( K ) into C 0 ( Γ, X ) spaces depend on the height of K? , J. Math. Anal. Appl. 402 (2013), 185–190. L. Candido, E. M. Galego How far C 0 ( Γ, X ) with Γ discrete from C 0 ( K , X ) spaces? , Fund. Math. 218(2012), 151–163. Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Banach 1932 T : c → c 0 T ( x 1 , x 2 , x 3 , ... ) = (2 n →∞ x n , x 1 − lim lim n →∞ x n , x 2 − lim n →∞ x n , ... ) . Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Banach 1932 T : c → c 0 T ( x 1 , x 2 , x 3 , ... ) = (2 n →∞ x n , x 1 − lim lim n →∞ x n , x 2 − lim n →∞ x n , ... ) . � T � � T − 1 � = 3 . Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Banach 1932 T : c → c 0 T ( x 1 , x 2 , x 3 , ... ) = (2 n →∞ x n , x 1 − lim lim n →∞ x n , x 2 − lim n →∞ x n , ... ) . � T � � T − 1 � = 3 . � � � T �� T − 1 � : T is an isomorphism of X onto Y d ( X , Y ) = inf . T Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Banach 1932 T : c → c 0 T ( x 1 , x 2 , x 3 , ... ) = (2 n →∞ x n , x 1 − lim lim n →∞ x n , x 2 − lim n →∞ x n , ... ) . � T � � T − 1 � = 3 . � � � T �� T − 1 � : T is an isomorphism of X onto Y d ( X , Y ) = inf . T Cambern 1968 d ( c , c 0 ) = 3 . Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces d ( C ( K ) , c 0 ) =? Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces d ( C ( K ) , c 0 ) =? ⇒ K ≈ [1 , ω n k ] , com 1 ≤ n , k < ω. C ( K ) ∼ c 0 = Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces d ( C ( K ) , c 0 ) =? ⇒ K ≈ [1 , ω n k ] , com 1 ≤ n , k < ω. C ( K ) ∼ c 0 = Question d ( C ([1 , ω n k ]) , c 0 ) =? , for 1 ≤ n , k < ω. Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Definition A Banach space X � = { 0 } is said to have finite cotype 2 ≤ q < ∞ if there is a constant κ > 0 such that no matter how we select finitely many vectors v 1 , v 2 . . . , v n from X , � 1 �� 1 n n 2 � 1 � � v i � q ) q ≤ κ r i ( t ) v i � 2 dt ( � , 0 i =1 i =1 where r i : [0 , 1] → R denote the Rademacher functions , defined by setting r i ( t ) = sign (sin 2 i π t ) . Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Recall that the derivative of a topological space K is the space K (1) obtained by deleting from K its isolated points. The α -th derivative K ( α ) is defined recursively setting K (0) = K and � ( K ( δ ) ) (1) if α = δ + 1 , K ( α ) = � β<α K ( β ) if α is a limit ordinal . Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Recall that the derivative of a topological space K is the space K (1) obtained by deleting from K its isolated points. The α -th derivative K ( α ) is defined recursively setting K (0) = K and � ( K ( δ ) ) (1) if α = δ + 1 , K ( α ) = � β<α K ( β ) if α is a limit ordinal . Definition A topological space K is said to be scattered if K ( α ) = ∅ for some ordinal α . In this case, the minimal α such that K ( α ) = ∅ is called the height of K (in short, ht ( K )). Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Theorem Let K be a locally compact Hausdorff space, Γ an infinite set with the discrete topology and X a Banach space with finite cotype. Then for every integer n ≥ 1 and for every linear embedding T from C 0 ( K ) into C 0 ( Γ, X ) we have K ( n ) � = ∅ = ⇒ � T � � T − 1 � ≥ 2 n + 1 . Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces • For a non-empty closed subset K 1 ⊆ K we denote � f � K 1 = sup {| f ( x ) |} . x ∈ K 1 • For every function f ∈ C 0 ( K , X ) and ǫ > 0 we denote K ( f , ǫ ) = { x ∈ K : � f ( x ) � ≥ ǫ } . • For n + 1 functions g 0 , g 1 , . . . , g n in C 0 ( K ) satisfying 0 ≤ g 0 ( x ) ≤ g 1 ( x ) ≤ . . . ≤ g n ( x ) ≤ 1 , ∀ x ∈ K , we denote by F g 0 ,..., g n the set of all ( f 1 , . . . , f n ) ∈ C 0 ( K ) n such that 0 ≤ g 0 ( x ) ≤ f 1 ( x ) ≤ g 1 ( x ) ≤ . . . ≤ f n ( x ) ≤ g n ( x ) , ∀ x ∈ K . Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Proposition Let J and K be locally compact Hausdorff spaces, X a Banach space with finite cotype and suppose that T is a linear embedding of C 0 ( K ) into C 0 ( J , X ) with � T − 1 � = 1 and � T � < 2 n + 1 for some integer n ≥ 1. Take δ > 0 and θ < 1 such that � T � + 2 δ ≤ (2 n + 1) θ , and g 0 , g 1 , . . . , g n in C 0 ( K ) satisfying 0 ≤ g 0 ( x ) ≤ g 1 ( x ) ≤ . . . ≤ g n ( x ) ≤ 1 , ∀ x ∈ K . Assume that for each 1 ≤ i < j ≤ n K ( Tg i , δ 2 n ) ∩ K ( Tg j , δ 2 n ) = ∅ . Then n � � f i ) , δ ) ∩ J (1) � = ∅ . � g 0 � K (1) > θ = ⇒ K ( T ( F g 0 ,..., gn i =1 Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Theorem Let K be a locally compact Hausdorff space, Γ an infinite set with the discrete topology and X a Banach space with finite cotype. Suppose that there exists a linear embedding T from from C 0 ( K ) into C 0 ( Γ, X ). Then K has finite height and � T � � T − 1 � ≥ 2 ht ( K ) − 1 . Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces C ([0 , 1]) ֒ → C 0 ( N , C ([0 , 1])) . Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces C ([0 , 1]) ֒ → C 0 ( N , C ([0 , 1])) . [0 , 1] ( ω ) = [0 , 1] . Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Corollary Let X a Banach space with finite cotype and 1 ≤ n , k < ω . Then d ( C ([1 , ω n k ] , X ) , C 0 ( N , X )) ≥ 2 n + 1 . Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Recall that every ordinal number 1 ≤ ξ < ω ω has an unique representation in the Cantor normal form , ξ = ω n k m k + . . . + ω n 2 m 2 + ω n 1 m 1 where 0 ≤ n 1 < n 2 < . . . < n k < ω and 1 ≤ m 1 < ω , 1 ≤ m 2 < ω, . . . , 1 ≤ m k < ω and 1 ≤ k < ω . Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Recall that every ordinal number 1 ≤ ξ < ω ω has an unique representation in the Cantor normal form , ξ = ω n k m k + . . . + ω n 2 m 2 + ω n 1 m 1 where 0 ≤ n 1 < n 2 < . . . < n k < ω and 1 ≤ m 1 < ω , 1 ≤ m 2 < ω, . . . , 1 ≤ m k < ω and 1 ≤ k < ω . Definition For an ordinal number 1 ≤ ξ < ω ω , represented in the Cantor normal form as above, we set ξ [0] = ξ and by induction � ω n k m k + . . . + ω n 2 m 2 + ω n 1 +1 if r = 1 , ξ [ r ] = � ξ [ r − 1] � [1] if 1 ≤ r < ω. Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Let Γ n be the ordinal space [1 , ω n ] provided with the discrete topology and replace the space C 0 ( N , X ) by C 0 ( Γ n , X ). Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces

Bibliography Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces Let Γ n be the ordinal space [1 , ω n ] provided with the discrete topology and replace the space C 0 ( N , X ) by C 0 ( Γ n , X ). For each function f ∈ C ([1 , ω n ] , X ) set T ( f ) : Γ n → X by � 2 f ( ω n ) if ξ = ω n , T ( f )( ξ ) = f ( ξ ) − f ( ξ [1] ) if 1 ≤ ξ < ω n . Embeddings of C 0 ( K ) into C 0 ( Γ, X ) spaces