Free spaces and the approximation property

Transcription

1 Free spaces and the approximation property Gilles Godefroy Institut de Mathématiques de Jussieu Paris Rive Gauche (CNRS & UPMC-Université Paris-06) September 11, 2015 Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property -1 / 19

2 1. Free spaces 1 1. Free spaces 2 2. Approximation properties The main result Problems. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 0 / 19

3 1. Free spaces Basic definitions Definition Let M be a metric space equipped with a distinguished point 0 M. The space Lip 0 (M) of real-valued Lipschitz functions which vanish at 0 M is a Banach space for the Lipschitz norm, and its natural predual, i.e. the closed linear span of the Dirac measures, is denoted by F(M) and is called the Lipschitz-free space over M. Notation The distinguished point 0 M is a matter of convenience and changing it does not alter the isometric structure of the spaces we consider. Hence we will omit it and use the notation Lip(M) (resp. Lip(M, X )) for real-valued (resp. X-valued with X a Banach space) Lipschitz functions on M, always assuming that these functions vanish at 0 M. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 1 / 19

4 1. Free spaces Basic definitions Definition Let M be a metric space equipped with a distinguished point 0 M. The space Lip 0 (M) of real-valued Lipschitz functions which vanish at 0 M is a Banach space for the Lipschitz norm, and its natural predual, i.e. the closed linear span of the Dirac measures, is denoted by F(M) and is called the Lipschitz-free space over M. Notation The distinguished point 0 M is a matter of convenience and changing it does not alter the isometric structure of the spaces we consider. Hence we will omit it and use the notation Lip(M) (resp. Lip(M, X )) for real-valued (resp. X-valued with X a Banach space) Lipschitz functions on M, always assuming that these functions vanish at 0 M. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 1 / 19

5 1. Free spaces Extension of Lipschitz functions Let A be a metric space, and B be a non-empty subset of A. It is well-known that real-valued Lipschitz functions on B can be extended to Lipschitz functions on A with the same Lipschitz constant with an inf-convolution formula. Namely, if f : B R is L-Lipschitz, then the formula f (a) = inf{f (b) + Ld(a, b); b B} which goes back to Mac Shane (1934) defines a L-Lipschitz function f on A which extends f. This formula relies of course on the order structure of the real line. Therefore it cannot be used for extending Banach-space valued Lipschitz functions - except in some specific cases. Another drawback of this formula is that it is not linear in f, that is, the map f f that it defines is not a linear one. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 2 / 19

6 1. Free spaces It is known that Banach-space valued Lipschitz functions do not generally admit Lipschitz extensions (Lindenstrauss, 1964), and that generally no continuous linear extension operator exists for Lipschitz real-valued functions. Canonical examples will be shown below. We will relate these two conditions (extending Banach-space valued functions, finding linear extension operators for real-valued Lipschitz functions) and a combination of both with the validity of Grothendieck s bounded approximation property for the Banach spaces which naturally show up in this context, namely the Lipschitz-free spaces. We will be dealing with finite subsets of a compact metric space M which approximate this space M, and these extension properties are easy for any given finite subset, but what matters is to find a uniform bound on the norm of the extension operators for this approximating sequence. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 3 / 19

7 1. Free spaces It is known that Banach-space valued Lipschitz functions do not generally admit Lipschitz extensions (Lindenstrauss, 1964), and that generally no continuous linear extension operator exists for Lipschitz real-valued functions. Canonical examples will be shown below. We will relate these two conditions (extending Banach-space valued functions, finding linear extension operators for real-valued Lipschitz functions) and a combination of both with the validity of Grothendieck s bounded approximation property for the Banach spaces which naturally show up in this context, namely the Lipschitz-free spaces. We will be dealing with finite subsets of a compact metric space M which approximate this space M, and these extension properties are easy for any given finite subset, but what matters is to find a uniform bound on the norm of the extension operators for this approximating sequence. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 3 / 19

8 1. Free spaces It turns out that the extension condition should be relaxed: what matters eventually is not to design an exact extension of a function f defined on a subset S of M, but a function on M whose restriction to S is uniformly close to f. The proofs are simple, and rely on canonical constructions. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 4 / 19

9 2. Approximation properties Free spaces 2 2. Approximation properties The main result Problems. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 5 / 19

10 2. Approximation properties. We recall that a separable Banach space X has the bounded approximation property (in short, the B.A.P.) if there exists a sequence of finite rank operators T n such that lim T n (x) x = 0 for every x X. It follows then from the uniform boundedness principle that sup T n = λ < and then we say that X has the λ B.A.P. The B.A.P. and the existence of Schauder bases for Lipschitz-free spaces has already been investigated in a number of recent articles (by M. Cuth, A. Dalet, M. Doucha, P. Hajek, N. Kalton, G. Lancien, E. Pernecka, R. Smith). There exist (N. Ozawa, G. 2014) compact metric spaces K such that F(K) fails the approximation property (in short, A.P.): actually, if X is a separable Banach space failing the A.P. and C is a compact convex set containing 0 which spans a dense linear subspace of X, then F(C) fails the A.P. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 6 / 19

11 2. Approximation properties. We recall that a separable Banach space X has the bounded approximation property (in short, the B.A.P.) if there exists a sequence of finite rank operators T n such that lim T n (x) x = 0 for every x X. It follows then from the uniform boundedness principle that sup T n = λ < and then we say that X has the λ B.A.P. The B.A.P. and the existence of Schauder bases for Lipschitz-free spaces has already been investigated in a number of recent articles (by M. Cuth, A. Dalet, M. Doucha, P. Hajek, N. Kalton, G. Lancien, E. Pernecka, R. Smith). There exist (N. Ozawa, G. 2014) compact metric spaces K such that F(K) fails the approximation property (in short, A.P.): actually, if X is a separable Banach space failing the A.P. and C is a compact convex set containing 0 which spans a dense linear subspace of X, then F(C) fails the A.P. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 6 / 19

12 2. Approximation properties. Notation If K is a compact metric space and T : Lip(K) Lip(K) is a continuous linear operator, we denote by T L its operator norm when Lip(K) is equipped with the Lipschitz norm, and by T L, its norm when the domain space is equipped with the Lipschitz norm and the range space with the uniform norm - alternatively, T L, is the norm of T from Lip(K) to C(K) when these spaces are equipped with their canonical norms. We use the same notation for X -valued Lipschitz functions. It should be noted that if M is a metric compact space, then the uniform norm induces on the unit ball of Lip(M) the weak* topology associated with the free space F(M). Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 7 / 19

13 3. The main result Free spaces 2 2. Approximation properties The main result Problems. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 8 / 19

14 3. The main result. Dirac maps A subset S of a metric space M is said to be ɛ-dense if for all m M, one has inf{d(m, s); s S} ɛ. We recall that if M is an arbitrary metric space, then the Dirac map δ : M F(M) is an isometry. Let M be a compact metric space. Let (M n ) n be a sequence of finite ɛ n -dense subsets of M, with lim(ɛ n ) = 0. We denote by δ n : M n F(M n ) the Dirac map relative to M n. We denote by R n (f ) the restriction to M n of a function f defined on M. Let λ 1.With this notation, the following holds. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 9 / 19

15 3. The main result. Dirac maps A subset S of a metric space M is said to be ɛ-dense if for all m M, one has inf{d(m, s); s S} ɛ. We recall that if M is an arbitrary metric space, then the Dirac map δ : M F(M) is an isometry. Let M be a compact metric space. Let (M n ) n be a sequence of finite ɛ n -dense subsets of M, with lim(ɛ n ) = 0. We denote by δ n : M n F(M n ) the Dirac map relative to M n. We denote by R n (f ) the restriction to M n of a function f defined on M. Let λ 1.With this notation, the following holds. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 9 / 19

16 3. The main result. Dirac maps A subset S of a metric space M is said to be ɛ-dense if for all m M, one has inf{d(m, s); s S} ɛ. We recall that if M is an arbitrary metric space, then the Dirac map δ : M F(M) is an isometry. Let M be a compact metric space. Let (M n ) n be a sequence of finite ɛ n -dense subsets of M, with lim(ɛ n ) = 0. We denote by δ n : M n F(M n ) the Dirac map relative to M n. We denote by R n (f ) the restriction to M n of a function f defined on M. Let λ 1.With this notation, the following holds. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 9 / 19

17 3. The main result. Main result Theorem The following assertions are equivalent: (1) The free space F(M) over M has the λ BAP. (2) There exist α n 0 with lim α n = 0 such that for every Banach space X, there exist linear operators E n : Lip(M n, X ) Lip(M, X ) with E n L λ and R n E n I L, α n. (3) There exist linear operators G n : Lip(M n ) Lip(M) with G n L λ and lim R n G n I L, = 0. (4) For every Banach space X, there exist β n 0 with lim β n = 0 such that for every 1-Lipschitz function F : M n X, there exists a λ- Lipschitz function H : M X such that R n (H) F l (M n,x ) β n. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 10 / 19

18 3. The main result. Outline of proof The crucial implication is (1) (2): Let Z = c((f(m n )) be the Banach space of sequences (µ n ) with µ n F(M n ) for all n, such that (µ n ) is norm-convergent in the Banach space F(M). We equip Z with the supremum norm, and we denote Q : Z F(M) the canonical quotient operator wich maps every sequence in Z to its limit. The kernel Z 0 = c 0 ((F(M n )) of Q is an M-ideal in Z, and the quotient space Z/Z 0 is isometric to F(M). It follows from (1) and the Ando-Choi-Effros theorem that there exists a linear map L : F(M) Z such that QL = Id F(M) and L λ. We let π n be the canonical projection from Z onto F(M n ), and we define g n = π n Lδ : M F(M n ). The maps g n are λ-lipschitz, and for every m M, we have lim g n (m) δ(m) F(M) = 0. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 11 / 19

19 3. The main result. Outline of proof The crucial implication is (1) (2): Let Z = c((f(m n )) be the Banach space of sequences (µ n ) with µ n F(M n ) for all n, such that (µ n ) is norm-convergent in the Banach space F(M). We equip Z with the supremum norm, and we denote Q : Z F(M) the canonical quotient operator wich maps every sequence in Z to its limit. The kernel Z 0 = c 0 ((F(M n )) of Q is an M-ideal in Z, and the quotient space Z/Z 0 is isometric to F(M). It follows from (1) and the Ando-Choi-Effros theorem that there exists a linear map L : F(M) Z such that QL = Id F(M) and L λ. We let π n be the canonical projection from Z onto F(M n ), and we define g n = π n Lδ : M F(M n ). The maps g n are λ-lipschitz, and for every m M, we have lim g n (m) δ(m) F(M) = 0. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 11 / 19

20 3. The main result. Outline of proof The crucial implication is (1) (2): Let Z = c((f(m n )) be the Banach space of sequences (µ n ) with µ n F(M n ) for all n, such that (µ n ) is norm-convergent in the Banach space F(M). We equip Z with the supremum norm, and we denote Q : Z F(M) the canonical quotient operator wich maps every sequence in Z to its limit. The kernel Z 0 = c 0 ((F(M n )) of Q is an M-ideal in Z, and the quotient space Z/Z 0 is isometric to F(M). It follows from (1) and the Ando-Choi-Effros theorem that there exists a linear map L : F(M) Z such that QL = Id F(M) and L λ. We let π n be the canonical projection from Z onto F(M n ), and we define g n = π n Lδ : M F(M n ). The maps g n are λ-lipschitz, and for every m M, we have lim g n (m) δ(m) F(M) = 0. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 11 / 19

21 3. The main result. Outline of proof, continued Since M is compact, this implies by an equicontinuity argument that if we let then lim α n = 0. α n = sup g n (m) δ(m) F(M) m M Let now X be a Banach space, and F : M n X be a Lipschitz map. There exists a unique continuous linear map F : F(M n ) X such that F δ n = F, and its norm is equal to the Lipschitz constant of F. In the notation of (Kalton-G. 2003), one has F = β X ˆF and in particular F depends linearly upon F. We let now E n (F ) = F g n and it is easy to check that the sequence (E n ) satisfies the requirements of (2). Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 12 / 19

22 3. The main result. Outline of proof, continued Since M is compact, this implies by an equicontinuity argument that if we let then lim α n = 0. α n = sup g n (m) δ(m) F(M) m M Let now X be a Banach space, and F : M n X be a Lipschitz map. There exists a unique continuous linear map F : F(M n ) X such that F δ n = F, and its norm is equal to the Lipschitz constant of F. In the notation of (Kalton-G. 2003), one has F = β X ˆF and in particular F depends linearly upon F. We let now E n (F ) = F g n and it is easy to check that the sequence (E n ) satisfies the requirements of (2). Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 12 / 19

23 3. The main result. Examples. Let M be a compact metric space with distinguished point 0 M, such that F(M) fails the B.A.P. (such an M exists by Ozawa-G.). We denote by M the Cartesian product of countably many copies of M equipped with d (x n, y n ) = sup d(x n, y n ), and by P n : M M the corresponding sequence of projections. We let X = l (F(M n )). We define a map from the subset L = Π n 1 M n of M to X by the formula ((m n )) = ( δ n (m n )) n. The map is 1-Lipschitz, and it cannot be extended to a Lipschitz map from M to X. Moreover, there does not exist a linear extension operator E : Lip(L) Lip(M ) with E L <. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 13 / 19

24 3. The main result. Examples. Let M be a compact metric space with distinguished point 0 M, such that F(M) fails the B.A.P. (such an M exists by Ozawa-G.). We denote by M the Cartesian product of countably many copies of M equipped with d (x n, y n ) = sup d(x n, y n ), and by P n : M M the corresponding sequence of projections. We let X = l (F(M n )). We define a map from the subset L = Π n 1 M n of M to X by the formula ((m n )) = ( δ n (m n )) n. The map is 1-Lipschitz, and it cannot be extended to a Lipschitz map from M to X. Moreover, there does not exist a linear extension operator E : Lip(L) Lip(M ) with E L <. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 13 / 19

25 4. Problems Free spaces 2 2. Approximation properties The main result Problems. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 14 / 19

26 4. Problems. Problem 1 The following question has been asked by Nigel Kalton. A positive answer would bear important consequences, a negative answer as well. Let D be a uniformly discrete metric space. Does the space F(D) have the bounded approximation property? Nigel Kalton showed in particular that this space has the Approximation Property. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 15 / 19

27 4. Problems. Problem 2 Let M be a compact subset of R n equipped with the Euclidean distance. Does the space F(M) have the Metric Approximation Property? G. Lancien and E. Pernecka showed that if M is a doubling metric space, then F(M) has the Bounded Approximation Property. It follows from their proof that a subset of R n has the Bounded Approximation Property with a constant controlled by n. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 16 / 19

28 4. Problems. Problem 3 Our next problem is formulated in a rather informal way. The existence of Banach spaces failing the Approximation Property shows the existence of nested metric spaces such that bounded linear extension operators for Lipschitz functions fail to exist. It is natural to wonder if the connection provided by our main result can work the other way. That is: Is it possible to use natural examples from the theory of metric spaces, in order to construct Lipschitz-free Banach spaces which fail the Approximation Property? Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 17 / 19

29 4. Problems. Problem 4 The construction from Ozawa-G. shows that there is a distance d 0 on the Hilbert cube H = [0, 1] N which induces the product topology and which is such that the corresponding free space F((H, d 0 )) fails the A. P. The set H of distances on H which induce the product topology in a Polish space (and thus a Baire space) when equipped with the uniform convergence on H H. In this frame, we can ask Is the subset of H consisting of all distances d such that F((H, d)) fails the Approximation Property residual in H? If it is so, there is some hope to use the Baire category lemma to obtain simpler proofs of the existence of Banach spaces failing the Approximation Property. Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 18 / 19

30 4. Problems. GOOD LUCK RAFA! Gilles Godefroy (IMJ-PRG) Free spaces and the approximation property 19 / 19