Supports and approximation properties in Lipschitz-free spaces Eva - PowerPoint PPT Presentation

Lipschitz-free spaces Supports Approximation properties Supports and approximation properties in Lipschitz-free spaces Eva Perneck´ a Czech Technical University, Prague Workshop on Banach spaces and Banach lattices Madrid, September 2019 Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 1 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure Let ( M , d ) and ( N , ̺ ) be metric spaces. A map f : M − → N is called Lipschitz if there exists a constant C > 0 such that ̺ ( f ( p ) , f ( q )) ≤ C d ( p , q ) ∀ p , q ∈ M . The Lipschitz constant of f is defined as � ̺ ( f ( p ) , f ( q )) � Lip( f ) := sup : p , q ∈ M , p � = q . d ( p , q ) Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 2 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure Let ( M , d ) and ( N , ̺ ) be metric spaces. A map f : M − → N is called Lipschitz if there exists a constant C > 0 such that ̺ ( f ( p ) , f ( q )) ≤ C d ( p , q ) ∀ p , q ∈ M . The Lipschitz constant of f is defined as � ̺ ( f ( p ) , f ( q )) � Lip( f ) := sup : p , q ∈ M , p � = q . d ( p , q ) Theorem (McShane, ’34) Let S ⊆ M. Then every Lipschitz function f : S − → R can be extended to a Lipschitz function � → R so that Lip( � f : M − f ) = Lip( f ) . � f ( p ) := sup { f ( q ) − Lip( f ) d ( p , q ) : q ∈ S } Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 2 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure Let ( M , d ) be a complete metric space with a base point 0 ∈ M (called a pointed metric space ). The Lipschitz-free space over M , denoted F ( M ) , is the Banach space satisfying the following universal property: There exists an isometric embedding δ : M − → F ( M ) such that span δ ( M ) = F ( M ) and δ (0) = 0. For any Banach space X and any Lipschitz map L : M − → X with L (0) = 0 there exists a unique linear operator ¯ L : F ( M ) − → X such that � ¯ L � = Lip( L ) and ¯ L δ = L , i.e. the following diagram commutes: L M X ¯ δ L F ( M ) Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 3 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure L M N ∀ M , N metric spaces, ∀ L Lipschitz with L (0) = 0 δ M δ N ∃ ! ˆ L linear operator s.t. � ˆ L � = Lip( L ) and ˆ L δ M = δ N L . ˆ L F ( M ) F ( N ) Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 4 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure L M N ∀ M , N metric spaces, ∀ L Lipschitz with L (0) = 0 δ M δ N ∃ ! ˆ L linear operator s.t. � ˆ L � = Lip( L ) and ˆ L δ M = δ N L . ˆ L F ( M ) F ( N ) δ N L F ( N ) M N Indeed, by universal property define δ M ˆ L := δ N L . δ N L F ( M ) Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 4 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure L M N ∀ M , N metric spaces, ∀ L Lipschitz with L (0) = 0 δ M δ N ∃ ! ˆ L linear operator s.t. � ˆ L � = Lip( L ) and ˆ L δ M = δ N L . ˆ L F ( M ) F ( N ) δ N L F ( N ) M N Indeed, by universal property define δ M ˆ L := δ N L . δ N L F ( M ) If M and N are bi-Lipschitz homeomorphic, then F ( M ) and F ( N ) are linearly isomorphic. If M and N are isometric, then F ( M ) and F ( N ) are linearly isometric. Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 4 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure F ( c 0 ) is linearly isomorphic to F ( C ([0 , 1])). ( Dutrieux, Ferenczi, ’05 ) F ( B R n ) is linearly isomorphic to F ( R n ). ( Kaufmann, ’15 ) There exist ( K α ) α<ω 1 homeomorphic to the Cantor space such that F ( K α ) is not linearly isomorphic to F ( K β ). ( H´ ajek, Lancien, P, ’16 ) Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 5 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure Theorem (Godefroy, Kalton, ’03) A construction of examples of non-separable Banach spaces which are bi-Lipschitz homeomorphic but not linearly isomorphic. Theorem (Godefroy, Kalton, ’03) If a separable Banach space X is isometric to a subset of a Banach space Y , then X is already linearly isometric to a subspace of Y . Theorem (Godefroy, Kalton, ’03) Let X be a Banach space with the bounded approximation property. If a Banach space Y is bi-Lipschitz homeomorphic to X, then Y also has the bounded approximation property. Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 6 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure Let ( M , d ) be a complete pointed metric space with the base point 0 ∈ M . Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 7 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure Let ( M , d ) be a complete pointed metric space with the base point 0 ∈ M . Space of Lipschitz functions Then Lip 0 ( M ) = { f : M − → R : f Lipschitz , f (0) = 0 } with the norm � f � = Lip( f ) is a Banach space. Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 7 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure Let ( M , d ) be a complete pointed metric space with the base point 0 ∈ M . Space of Lipschitz functions Then Lip 0 ( M ) = { f : M − → R : f Lipschitz , f (0) = 0 } with the norm � f � = Lip( f ) is a Banach space. For p ∈ M consider the evaluation functional δ ( p ) ∈ Lip 0 ( M ) ∗ defined by � f , δ ( p ) � = f ( p ) ∀ f ∈ Lip 0 ( M ) . Then the Dirac map δ : M → Lip 0 ( M ) ∗ is an isometric embedding. Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 7 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure Let ( M , d ) be a complete pointed metric space with the base point 0 ∈ M . Space of Lipschitz functions Then Lip 0 ( M ) = { f : M − → R : f Lipschitz , f (0) = 0 } with the norm � f � = Lip( f ) is a Banach space. For p ∈ M consider the evaluation functional δ ( p ) ∈ Lip 0 ( M ) ∗ defined by � f , δ ( p ) � = f ( p ) ∀ f ∈ Lip 0 ( M ) . Then the Dirac map δ : M → Lip 0 ( M ) ∗ is an isometric embedding. Lipschitz-free space The space F ( M ) = span �·� δ ( M ) ⊆ Lip 0 ( M ) ∗ with the norm inherited from Lip 0 ( M ) ∗ is the Lipschitz-free space over M . Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 7 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure F ( M ) ∗ ≡ Lip 0 ( M ) and for ( f γ ) and f in B Lip 0 ( M ) we have w ∗ f γ − − → f ⇐ ⇒ ( f γ ( p ) − → f ( p ) ∀ p ∈ M ) . Theorem (Weaver, ’17) If M has a finite diameter or it is complete and convex (e.g. Banach space) then F ( M ) is the unique predual of Lip 0 ( M ) . Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 8 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure F ( M ) ∗ ≡ Lip 0 ( M ) and for ( f γ ) and f in B Lip 0 ( M ) we have w ∗ f γ − − → f ⇐ ⇒ ( f γ ( p ) − → f ( p ) ∀ p ∈ M ) . Theorem (Weaver, ’17) If M has a finite diameter or it is complete and convex (e.g. Banach space) then F ( M ) is the unique predual of Lip 0 ( M ) . Theorem (Kadets, ’85) If K is a subset of M containing the base point, then F ( K ) is isometric to a subspace of F ( M ) . Precisely, F ( K ) ≡ F M ( K ) := span δ ( K ) ⊆ F ( M ) . If K is a Lipschitz retract of M , then F M ( K ) is complemented in F ( M ). Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 8 / 41

Lipschitz-free spaces Universal property Supports Construction Approximation properties Linear structure F ( M ) ∗ ≡ Lip 0 ( M ) and for ( f γ ) and f in B Lip 0 ( M ) we have w ∗ f γ − − → f ⇐ ⇒ ( f γ ( p ) − → f ( p ) ∀ p ∈ M ) . Theorem (Weaver, ’17) If M has a finite diameter or it is complete and convex (e.g. Banach space) then F ( M ) is the unique predual of Lip 0 ( M ) . Theorem (Kadets, ’85) If K is a subset of M containing the base point, then F ( K ) is isometric to a subspace of F ( M ) . Precisely, F ( K ) ≡ F M ( K ) := span δ ( K ) ⊆ F ( M ) . If K is a Lipschitz retract of M , then F M ( K ) is complemented in F ( M ). For a closed subset K ⊆ M , define the kernel of K as I M ( K ) = { f ∈ Lip 0 ( M ) : f ( p ) = 0 ∀ p ∈ K } . Then F M ( K ) ⊥ = I M ( K ) and I M ( K ) ⊥ = F M ( K ). Eva Perneck´ a Supports and approximation properties in Lipschitz-free spaces 8 / 41