Extension of continuous convex functions II

Transcription

1 Interactions between Logic, Topological structures and Banach spaces theory May 19-24, 2013 Eilat Joint work with Libor Veselý

2 Notations and preliminaries X real topological vector space. X topological dual of X endowed with the w -topology. X locally convex. Y subspace of X. Definition (X, Y ) has the CE-property if each continuous convex function f : Y R admits a continuous convex extension F : X R.

3 Notation (a) All nets of the form {t n,γ } n N,γ Γ, where Γ is a nonempty set, are directed by the following relation: (n 1, γ 1 ) (n 2, γ 2 ) iff n 1 n 2.

4 Notation (a) All nets of the form {t n,γ } n N,γ Γ, where Γ is a nonempty set, are directed by the following relation: (n 1, γ 1 ) (n 2, γ 2 ) iff n 1 n 2. (b) We denote by q the linear w -w -continuous map q : X Y, q(x ) = x Y.

5 Notation (a) All nets of the form {t n,γ } n N,γ Γ, where Γ is a nonempty set, are directed by the following relation: (n 1, γ 1 ) (n 2, γ 2 ) iff n 1 n 2. (b) We denote by q the linear w -w -continuous map q : X Y, q(x ) = x Y. (c) Given A X, we denote by A X the polar set of A, i.e. the set A = {x X ; x x 1 x A}.

6 Theorem The following assertions are equivalent. (i) (X, Y ) has the CE-property. (ii) For every sequence {B n } of w -compact convex equicontinuous sets in Y such that B n {0}, there exists a sequence {E n } of w -compact convex equicontinuous sets in X such that E n {0} and such that q(e n ) = B n, for each n N. (iii) For every equicontinuous w -null net {y n,γ} Y, there exists an equicontinuous w -null net {x n,γ} X such that x n,γ Y = y n,γ, for each (n, γ) N Γ.

7 In the sequel, V X is a convex symmetric neighborhood of 0. Definition Let µ 1. We say that (X, Y ) has the µ-ce-property if for each V as above and for each sequence {B n } of w -compact convex sets contained in q(v ) = (V Y ) such that B n {0}, there exists a sequence {E n } of w -compact convex equicontinuous sets contained in µv such that E n {0} and such that q(e n ) = B n, for each n N.

8 Proposition The following assertions are equivalent. (i) (X, Y ) has the µ-ce-property. (ii) For each V as above and for every w -null net {y n,γ} q(v ), there exists a w -null net {x n,γ} µv such that y n,γ = q(x n,γ) for each (n, γ) N Γ.

9 The CE-property and extension of operators Observation There exists a canonical one-to-one correspondence T : X c 0 (l (Γ)) {x n,γ} (n,γ) N Γ (w -null) given by the formula x n,γx = [(Tx)(n)](γ). Moreover, we have the equivalence T (V ) B c0 (l (Γ)) {x n,γ} V.

10 Theorem The following assertions are equivalent. (i) The couple (X, Y ) has the µ-ce-property. (ii) Given V as above and a nonempty set Γ, every operator T 0 : Y c 0 (l (Γ)) such that T 0 (V Y ) B c0 (l (Γ)), can be extended to an operator T : X c 0 (l (Γ)) such that T (V ) µb c0 (l (Γ)). (iii) Given V as above and a λ-injective Banach space E, every operator T 0 : Y c 0 (E) such that T 0 (V Y ) B c0 (E) can be extended to an operator T : X c 0 (E) such that T (V ) λµb c0 (E). If X is a normed space and V = B X, (ii) says that T µ T 0.

11 Definition Given λ 1, a Banach space E is said to be λ-separably injective iff for each separable Banach space X and each subspace Y X, every operator T 0 : Y E can be extended to an operator T : X E such that T λ T 0. Fact (Avilés, Cábello Sanchez, Castillo, Gonzáles, Moreno, 2013) Suppose that E is λ-separably injective, X /Y is separable, and ε > 0, then every operator T 0 : Y E can be extended to an operator T : X E such that T (2 + λ + ε) T 0

12 Definition X is conditionally separable if for each neighborhood W of 0 there exists a countable set Q X such that X = Q + W. Theorem Assume that X /Y is conditionally separable. Then the couple (X, Y ) has the 2-CE-property.

13 Corollary Assume that X /Y is conditionally separable. (a) (Rosenthal) If E is a λ-injective Banach space, then every operator T 0 : Y c 0 (E) such that T 0 (V Y ) B c0 (E) can be extended to an operator T : X c 0 (E) such that T (V ) 2λB c0 (E). (b) (Sobczyk) Every operator T 0 : Y c 0 such that T 0 (V Y ) B c0 can be extended to an operator T : X c 0 such that T (V ) 2B c0. (c) (Rosenthal) If E is a λ-separably injective Banach space and ε > 0, then every operator T 0 : Y c 0 (E) such that T 0 (V Y ) B c0 (E) can be extended to an operator T : X c 0 (E) such that T (V ) (2λ ε)b c0 (E).

14 The CE-property in normed spaces Notation 1 X normed space. 2 Y subspace of X.

15 Proposition (X, Y ) has the CE-property, for each subspace Y of X, in the following cases. (a) X is separable. (b) X is isomorphic to a Hilbert space.

16 Theorem Let Z γ (γ Γ) be separable Banach spaces. (X, Y ) has the CE-property in the following cases. (a) X = c 0 (Γ, Z γ ) or X = l p (Γ, Z γ ) (1 < p < ). (b) X = l 1 (Γ, Z γ ) and Y is closed in the σ ( X, c 0 (Γ, Z γ ) ) -topology. (c) X /Y is isomorphic to l 1 (Γ)/W, where W is a w -closed subspace of l 1 (Γ) (with respect to c 0 (Γ)).

17 Proposition Let λ 1. Let us consider the following assertions. (i) There exists a function ϕ : B Y λb X such that: ϕ(0) = 0 and ϕ is w -w -continuous at 0; q ϕ = Id Y. (ii) There exists a function ϕ : Y X such that: ϕ(0) = 0 and ϕ is w -w -sequentially continuous at 0; q ϕ = Id Y ; ϕ(y ) λ y, for each y Y. (iii) The couple (X, Y ) has the CE-property. Then (i) (ii) (iii). In the case X is separable, (i) holds with λ = 2.

18 Case X = c 0 (Γ, Z γ ). Notation Let Γ 0 Γ. We denote by R Γ0 : c 0 (Γ, Z γ ) c 0 (Γ, Z γ ) the canonical projection defined by { x(γ) if γ Γ 0 ; [R Γ0 (x)](γ) = 0 otherwise.

19 Lemma (Johnson and Zippin, 1989) Let X = c 0 (Γ). Then Γ can be decomposed into a family {Γ α } α A of pairwise disjoint countable sets such that Y α := R Γα (Y ) Y, for each α A.

20 Lemma Let X = c 0 (Γ, Z γ ). Then Γ can be decomposed into a family {Γ α } α A of pairwise disjoint countable sets such that Y α := R Γα (Y ) Y, for each α A.

21 proof of the theorem. Let {Γ α } α A and {Y α } α A be the decomposition guaranteed by the decomposition lemma. For every α A, let X α = R Γα X = c 0 (Γ α, Z γ ). Let ϕ α : Y α X α be such that: ϕ α (0) = 0 and ϕ α is w -w -sequentially continuous at 0; q α ϕ α = Id Y α, where q α : X α Y α is the canonical restriction map; ϕ α (y ) 2 y, for each y Y α. Define ϕ : Y X as follows: let γ Γ and let α A be such that γ Γ α, define ϕ(y )(γ) = ϕ α (y Yα )(γ).

22 Theorem Let Z γ (γ Γ) be separable Banach spaces. (X, Y ) has the CE(BX 0 )-property in the following cases. (a) X = c 0 (Γ, Z γ ) or X = l p (Γ, Z γ ) (1 < p < ). (b) X = l 1 (Γ, Z γ ) and Y is closed in the σ ( X, c 0 (Γ, Z γ ) ) -topology.

23 A. Avilés, F. Cábello Sanchez, J.M.F. Castillo, M. Gonzáles and Y. Moreno, On separably injective Banach spaces, Adv. in Math. 234 (2013), C.A. De Bernardi and L. Veselý, Extension of continuous convex functions from subspaces I, preprint, C.A. De Bernardi and L. Veselý, Extension of continuous convex functions from subspaces II, preprint, Johnson, W. B. and Zippin, M., Extension of operators from subspaces of c 0 (Γ) into C(K) spaces, Proc. Amer. Math. Soc. 107 (1989), H.P. Rosenthal, The complete separable extension property, J. Oper. Theory 43 (2000),

24 Thank you!