Non-trivial extensions of c 0 by reflexive spaces

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1 Non-trivial extensions of c 0 by reflexive spaces Tomasz Kochanek University of Silesia Institute of Mathematics Katowice, Poland The 12 th Debrecen-Katowice Winter Seminar on Functional Equations and Inequalities Hajdúszoboszló, Hungary January 25 28, 2012 Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 1 / 13

2 Basic definitions Exact sequences Let X, Y, Z be F -spaces. A short exact sequence is a diagram ( ) 0 Y i Z q X 0, where i : Y Z is a one-to-one operator with a closed range (embedding) and q : Z X is a surjective operator such that im(i) = ker(q). Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 2 / 13

3 Basic definitions Exact sequences Let X, Y, Z be F -spaces. A short exact sequence is a diagram ( ) 0 Y i Z q X 0, where i : Y Z is a one-to-one operator with a closed range (embedding) and q : Z X is a surjective operator such that im(i) = ker(q). In other words, Z contains a closed subspace Y 1 Y (isomorphically) such that the quotient space Z/Y 1 X. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 2 / 13

4 Basic definitions Exact sequences Let X, Y, Z be F -spaces. A short exact sequence is a diagram ( ) 0 Y i Z q X 0, where i : Y Z is a one-to-one operator with a closed range (embedding) and q : Z X is a surjective operator such that im(i) = ker(q). In other words, Z contains a closed subspace Y 1 Y (isomorphically) such that the quotient space Z/Y 1 X. We then say that Z is a twisted sum of Y and X (in this order!), or that Z is an extension of X by Y. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 2 / 13

5 Basic definitions Equivalent exact sequences In fact, twisted sums are identified via the following natural equivalence relation: Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 3 / 13

6 Basic definitions Equivalent exact sequences In fact, twisted sums are identified via the following natural equivalence relation: We say that two exact sequences of F -spaces 0 Y Z 1 X 0 and 0 Y Z 2 X 0 are equivalent, if there exists an operator T : Z 1 Z 2 such that the diagram is commutative. 0 Y Z 1 X 0 0 Y Z 2 X 0 T Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 3 / 13

7 Basic definitions Splitting of exact sequences For any two F -spaces X and Y we have always the trivial exact sequence: ( ) 0 Y Y X X 0 produced by the direct sum, jointly with the natural embedding and projection. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 4 / 13

8 Basic definitions Splitting of exact sequences For any two F -spaces X and Y we have always the trivial exact sequence: ( ) 0 Y Y X X 0 produced by the direct sum, jointly with the natural embedding and projection. We say that exact sequence ( ) splits if and only if it is equivalent to ( ). Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 4 / 13

9 Basic definitions Splitting of exact sequences For any two F -spaces X and Y we have always the trivial exact sequence: ( ) 0 Y Y X X 0 produced by the direct sum, jointly with the natural embedding and projection. We say that exact sequence ( ) splits if and only if it is equivalent to ( ). Equivalently: the copy i(y ) of Y, inside Z, is complemented in Z. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 4 / 13

10 Basic definitions Splitting of exact sequences For any two F -spaces X and Y we have always the trivial exact sequence: ( ) 0 Y Y X X 0 produced by the direct sum, jointly with the natural embedding and projection. We say that exact sequence ( ) splits if and only if it is equivalent to ( ). Equivalently: the copy i(y ) of Y, inside Z, is complemented in Z. In such a case we must have Z X Y. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 4 / 13

11 Basic definitions Functor Ext Now, we focus on the case where X and Y are Banach spaces. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 5 / 13

12 Basic definitions Functor Ext Now, we focus on the case where X and Y are Banach spaces. The functor Ext assigns, to every pair (X, Y ) of Banach spaces, the class of all locally convex twisted sums of Y and X, modulo the equivalence relation defined earlier. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 5 / 13

13 Basic definitions Functor Ext Now, we focus on the case where X and Y are Banach spaces. The functor Ext assigns, to every pair (X, Y ) of Banach spaces, the class of all locally convex twisted sums of Y and X, modulo the equivalence relation defined earlier. In other words, Ext(X, Y ) collects the class of all Banach spaces Z which produce an exact sequence of the form ( ) 0 Y Z X 0. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 5 / 13

14 Basic definitions Functor Ext Now, we focus on the case where X and Y are Banach spaces. The functor Ext assigns, to every pair (X, Y ) of Banach spaces, the class of all locally convex twisted sums of Y and X, modulo the equivalence relation defined earlier. In other words, Ext(X, Y ) collects the class of all Banach spaces Z which produce an exact sequence of the form ( ) 0 Y Z X 0. We write Ext(X, Y ) = 0 if every exact sequence ( ), where Z is a Banach space, splits. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 5 / 13

15 Some classical results if Y is a closed subspace of a Banach space X, and dimy < or codimy <, then Ext(X, Y ) = 0 (quite direct application of the Hahn-Banach theorem); Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 6 / 13

16 Some classical results if Y is a closed subspace of a Banach space X, and dimy < or codimy <, then Ext(X, Y ) = 0 (quite direct application of the Hahn-Banach theorem); Ext(X, l ) = 0 for every Banach space X (injectivity of l ); Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 6 / 13

17 Some classical results if Y is a closed subspace of a Banach space X, and dimy < or codimy <, then Ext(X, Y ) = 0 (quite direct application of the Hahn-Banach theorem); Ext(X, l ) = 0 for every Banach space X (injectivity of l ); Ext(X, c 0 ) = 0 for every separable Banach space (Sobczyk s theorem); Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 6 / 13

18 Some classical results if Y is a closed subspace of a Banach space X, and dimy < or codimy <, then Ext(X, Y ) = 0 (quite direct application of the Hahn-Banach theorem); Ext(X, l ) = 0 for every Banach space X (injectivity of l ); Ext(X, c 0 ) = 0 for every separable Banach space (Sobczyk s theorem); Ext(L 1 (µ), Y ) = 0 for every positive measure µ and every Banach space Y, complemented in its bidual (Lindenstrauss lifting principle); Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 6 / 13

19 Some classical results if Y is a closed subspace of a Banach space X, and dimy < or codimy <, then Ext(X, Y ) = 0 (quite direct application of the Hahn-Banach theorem); Ext(X, l ) = 0 for every Banach space X (injectivity of l ); Ext(X, c 0 ) = 0 for every separable Banach space (Sobczyk s theorem); Ext(L 1 (µ), Y ) = 0 for every positive measure µ and every Banach space Y, complemented in its bidual (Lindenstrauss lifting principle); Ext(l 2, l 2 ) 0 (Enflo & Lindenstrauss & Pisier, 1975 and, by a much more universal method, Kalton & Peck, 1979); Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 6 / 13

20 Some classical results if Y is a closed subspace of a Banach space X, and dimy < or codimy <, then Ext(X, Y ) = 0 (quite direct application of the Hahn-Banach theorem); Ext(X, l ) = 0 for every Banach space X (injectivity of l ); Ext(X, c 0 ) = 0 for every separable Banach space (Sobczyk s theorem); Ext(L 1 (µ), Y ) = 0 for every positive measure µ and every Banach space Y, complemented in its bidual (Lindenstrauss lifting principle); Ext(l 2, l 2 ) 0 (Enflo & Lindenstrauss & Pisier, 1975 and, by a much more universal method, Kalton & Peck, 1979); Ext(l p, l 1 ) 0 for each 1 p < (by the so-called push-out construction applied to the Kalton Peck sequence); Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 6 / 13

21 Extensions of c 0 F. Cabello Sánchez, J.M.F. Castillo, Uniform boundedness and twisted sums of Banach spaces, Houston J. Math. 30 (2004), Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 7 / 13

22 Extensions of c 0 F. Cabello Sánchez, J.M.F. Castillo, Uniform boundedness and twisted sums of Banach spaces, Houston J. Math. 30 (2004), Ext(c 0, l 2 ) 0 Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 7 / 13

23 Extensions of c 0 F. Cabello Sánchez, J.M.F. Castillo, Uniform boundedness and twisted sums of Banach spaces, Houston J. Math. 30 (2004), Ext(c 0, l 2 ) 0 This result was also established for some special Banach spaces X, in the place of l 2. We study the following statement: Ext(c 0, X ) 0 for every reflexive Banach space X Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 7 / 13

24 Extensions of c 0 F. Cabello Sánchez, J.M.F. Castillo, Uniform boundedness and twisted sums of Banach spaces, Houston J. Math. 30 (2004), Ext(c 0, l 2 ) 0 This result was also established for some special Banach spaces X, in the place of l 2. We study the following statement: Ext(c 0, X ) 0 for every reflexive Banach space X and reduce it to a concrete question concerning almost additive measures with values in l 2. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 7 / 13

25 General strategy To show that Ext(c 0, l 2 ) 0 for each reflexive Banach space X, we shall combine four main ingredients: Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 8 / 13

26 General strategy To show that Ext(c 0, l 2 ) 0 for each reflexive Banach space X, we shall combine four main ingredients: (1) to find what consequences would the negation Ext(c 0, X ) = 0 have in language of stability properties of vector measures with values in X ; Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 8 / 13

27 General strategy To show that Ext(c 0, l 2 ) 0 for each reflexive Banach space X, we shall combine four main ingredients: (1) to find what consequences would the negation Ext(c 0, X ) = 0 have in language of stability properties of vector measures with values in X ; (2) to use Dvoretzky s theorem in order to transport almost additive l 2 -valued vector measures into almost additive X -valued vector measures; Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 8 / 13

28 General strategy To show that Ext(c 0, l 2 ) 0 for each reflexive Banach space X, we shall combine four main ingredients: (1) to find what consequences would the negation Ext(c 0, X ) = 0 have in language of stability properties of vector measures with values in X ; (2) to use Dvoretzky s theorem in order to transport almost additive l 2 -valued vector measures into almost additive X -valued vector measures; (3) to construct an appropriate creature: a certain almost additive l 2 -valued vector measure which would enjoy some special properties, just to make it possible... Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 8 / 13

29 General strategy To show that Ext(c 0, l 2 ) 0 for each reflexive Banach space X, we shall combine four main ingredients: (1) to find what consequences would the negation Ext(c 0, X ) = 0 have in language of stability properties of vector measures with values in X ; (2) to use Dvoretzky s theorem in order to transport almost additive l 2 -valued vector measures into almost additive X -valued vector measures; (3) to construct an appropriate creature: a certain almost additive l 2 -valued vector measure which would enjoy some special properties, just to make it possible... (4) to use the Diestel Faires theorem, for showing that the desire stability effect is impossible. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 8 / 13

30 General strategy To show that Ext(c 0, l 2 ) 0 for each reflexive Banach space X, we shall combine four main ingredients: (1) to find what consequences would the negation Ext(c 0, X ) = 0 have in language of stability properties of vector measures with values in X ; (2) to use Dvoretzky s theorem in order to transport almost additive l 2 -valued vector measures into almost additive X -valued vector measures; (3) to construct an appropriate creature: a certain almost additive l 2 -valued vector measure which would enjoy some special properties, just to make it possible... (4) to use the Diestel Faires theorem, for showing that the desire stability effect is impossible. That is the plan for the proof by contradiction. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 8 / 13

31 Stability for vector measures Definition We say that a Banach space X has the SVM (stability of vector measures) property if and only if there exists a real constant v(x ) < (depending only on X ) such that given any set Ω, any field F 2 Ω, and any mapping ν : F X satisfying ν(a B) ν(a) ν(b) 1 for A, B F, A B =, there is a vector measure µ: F X with ν(a) µ(a) v(x ) for A F. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 9 / 13

32 Stability for vector measures Definition We say that a Banach space X has the SVM (stability of vector measures) property if and only if there exists a real constant v(x ) < (depending only on X ) such that given any set Ω, any field F 2 Ω, and any mapping ν : F X satisfying ν(a B) ν(a) ν(b) 1 for A, B F, A B =, there is a vector measure µ: F X with ν(a) µ(a) v(x ) for A F. By the Kalton Roberts theorem (1983), the space R has the SVM property. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 9 / 13

33 Characterization of the SVM property Theorem Let X be a Banach space complemented in its bidual. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 10 / 13

34 Characterization of the SVM property Theorem Let X be a Banach space complemented in its bidual. Then the following assertions are equivalent: Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 10 / 13

35 Characterization of the SVM property Theorem Let X be a Banach space complemented in its bidual. Then the following assertions are equivalent: (i) X has the SVM property; Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 10 / 13

36 Characterization of the SVM property Theorem Let X be a Banach space complemented in its bidual. Then the following assertions are equivalent: (i) X has the SVM property; (ii) Ext(X, l 1 ) = 0; Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 10 / 13

37 Characterization of the SVM property Theorem Let X be a Banach space complemented in its bidual. Then the following assertions are equivalent: (i) X has the SVM property; (ii) Ext(X, l 1 ) = 0; (iii) Ext(l, X ) = 0; Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 10 / 13

38 Characterization of the SVM property Theorem Let X be a Banach space complemented in its bidual. Then the following assertions are equivalent: (i) X has the SVM property; (ii) Ext(X, l 1 ) = 0; (iii) Ext(l, X ) = 0; (iv) Ext(c 0, X ) = 0. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 10 / 13

39 Characterization of the SVM property Theorem Let X be a Banach space complemented in its bidual. Then the following assertions are equivalent: (i) X has the SVM property; (ii) Ext(X, l 1 ) = 0; (iii) Ext(l, X ) = 0; (iv) Ext(c 0, X ) = 0. Examples C(K) has the SVM property, for K being an extremally disconnected, compact, Hausdorff space; Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 10 / 13

40 Characterization of the SVM property Theorem Let X be a Banach space complemented in its bidual. Then the following assertions are equivalent: (i) X has the SVM property; (ii) Ext(X, l 1 ) = 0; (iii) Ext(l, X ) = 0; (iv) Ext(c 0, X ) = 0. Examples C(K) has the SVM property, for K being an extremally disconnected, compact, Hausdorff space; every von Neumann algebra has the SVM property; Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 10 / 13

41 Characterization of the SVM property Theorem Let X be a Banach space complemented in its bidual. Then the following assertions are equivalent: (i) X has the SVM property; (ii) Ext(X, l 1 ) = 0; (iii) Ext(l, X ) = 0; (iv) Ext(c 0, X ) = 0. Examples C(K) has the SVM property, for K being an extremally disconnected, compact, Hausdorff space; every von Neumann algebra has the SVM property; most of natural examples do not have the SVM property. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 10 / 13

42 The Dvoretzky theorem Theorem (Dvoretzky, 1961) Let X be an infinite-dimensional Banach space. Then for every ε > 0 and k N there is a subspace Y of X with dimy = k and d(l k 2, Y ) < 1 + ε. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 11 / 13

43 The Dvoretzky theorem Theorem (Dvoretzky, 1961) Let X be an infinite-dimensional Banach space. Then for every ε > 0 and k N there is a subspace Y of X with dimy = k and d(l k 2, Y ) < 1 + ε. Here d denotes the Banach-Mazur distance; let us recall that d(y, Z) = inf { T T 1 T : Y Z is an isomorphism }. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 11 / 13

44 The Dvoretzky theorem Theorem (Dvoretzky, 1961) Let X be an infinite-dimensional Banach space. Then for every ε > 0 and k N there is a subspace Y of X with dimy = k and d(l k 2, Y ) < 1 + ε. Here d denotes the Banach-Mazur distance; let us recall that d(y, Z) = inf { T T 1 T : Y Z is an isomorphism }. Another way to state Dvoretzky s theorem: for every ε > 0 and k N one may find a k-dimensional subspace Y X and an isomorphism T : l k 2 Y such that Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 11 / 13

45 The Dvoretzky theorem Theorem (Dvoretzky, 1961) Let X be an infinite-dimensional Banach space. Then for every ε > 0 and k N there is a subspace Y of X with dimy = k and d(l k 2, Y ) < 1 + ε. Here d denotes the Banach-Mazur distance; let us recall that d(y, Z) = inf { T T 1 T : Y Z is an isomorphism }. Another way to state Dvoretzky s theorem: for every ε > 0 and k N one may find a k-dimensional subspace Y X and an isomorphism T : l k 2 Y such that x T (x) (1 + ε) x for x l k 2. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 11 / 13

46 The Dvoretzky theorem Theorem (Dvoretzky, 1961) Let X be an infinite-dimensional Banach space. Then for every ε > 0 and k N there is a subspace Y of X with dimy = k and d(l k 2, Y ) < 1 + ε. Here d denotes the Banach-Mazur distance; let us recall that d(y, Z) = inf { T T 1 T : Y Z is an isomorphism }. Another way to state Dvoretzky s theorem: for every ε > 0 and k N one may find a k-dimensional subspace Y X and an isomorphism T : l k 2 Y such that x T (x) (1 + ε) x for x l k 2. Yet another way: l 2 is finitely represented in every infinite-dimensional Banach space. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 11 / 13

47 Diestel Faires theorem Theorem (Diestel & Faires, 1974) Let F be a field of sets and X be a Banach space. If there is a bounded, non-strongly additive vector measure µ: F X, then X contains an isomorphic copy of c 0. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 12 / 13

48 Diestel Faires theorem Theorem (Diestel & Faires, 1974) Let F be a field of sets and X be a Banach space. If there is a bounded, non-strongly additive vector measure µ: F X, then X contains an isomorphic copy of c 0. We say that µ: F X is strongly additive provided that the series n=1 µ(a n) is (unconditionally) norm convergent in X for every sequence (A n ) n=1 F of pairwise disjoint sets. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 12 / 13

49 Main result Let F = {A N : A < ℵ 0 or N \ A < ℵ 0 }. We wish to construct a function ν : F l 2 satisfying: Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 13 / 13

50 Main result Let F = {A N : A < ℵ 0 or N \ A < ℵ 0 }. We wish to construct a function ν : F l 2 satisfying: (i) ν is 1-additive, that is, for every A, B F with A B =, we have ν(a B) ν(a) ν(b) 1; Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 13 / 13

51 Main result Let F = {A N : A < ℵ 0 or N \ A < ℵ 0 }. We wish to construct a function ν : F l 2 satisfying: (i) ν is 1-additive, that is, for every A, B F with A B =, we have ν(a B) ν(a) ν(b) 1; (ii) for each n N we have ν{n} = Me n, where M is some fixed positive number and e n is the nth vector from the canonical basis of l 2 ; Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 13 / 13

52 Main result Let F = {A N : A < ℵ 0 or N \ A < ℵ 0 }. We wish to construct a function ν : F l 2 satisfying: (i) ν is 1-additive, that is, for every A, B F with A B =, we have ν(a B) ν(a) ν(b) 1; (ii) for each n N we have ν{n} = Me n, where M is some fixed positive number and e n is the nth vector from the canonical basis of l 2 ; (iii) ν is bounded. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 13 / 13

53 Main result Let F = {A N : A < ℵ 0 or N \ A < ℵ 0 }. We wish to construct a function ν : F l 2 satisfying: (i) ν is 1-additive, that is, for every A, B F with A B =, we have ν(a B) ν(a) ν(b) 1; (ii) for each n N we have ν{n} = Me n, where M is some fixed positive number and e n is the nth vector from the canonical basis of l 2 ; (iii) ν is bounded. Theorem Suppose that for arbitrarily large, but fixed, positive number M there is a function ν : F l 2 enjoying properties (i)-(iii). Then Ext(c 0, X ) 0 for every reflexive Banach space. Tomasz Kochanek (University of Silesia) Non-trivial extensions of c 0 13 / 13