on some applications of fra ss e theory to banach space...

On some applications of Fraısse theory to Banachspace theory

Michal Doucha

University of Franche-Comte, Besancon, France

July 19, 2016

Michal Doucha On some applications of Fraısse theory to Banach space theory

Fraısse theory

Fraısse theory is a collection of methods of constructing ‘universaland homogeneous’ structures.

Let X be a discrete structure. We say that X is homogeneous ifevery isomorphism between two finite (resp. finitely generated)substructures of X extends to an automorphism of X .

Let Y be a metric structure. We say that Y isalmost-homogeneous if every ‘almost-isometric’ isomorphismbetween two finite (resp. finitely generated) substructures of Yextends to an almost-isometric automorphism of Y .


Examples of homogeneous structures

The rational numbers (Q, <) with its linear order.

For every n ∈ N denote by Kn the n-complete graph. Thenthe disjoint union of countably many copies of Kn ishomogeneous.

Vector spaces with countable (algebraic) basis.

Finite fields.

Hilbert spaces.


Age of a structure

For a countable structure X , we denote by Age(X ) the set of allits finite (resp. finitely generated) substructures.

We observe that if X is homogeneous, then Age(X ) has thefollowing properties:

It is countable.

It is “hereditary”, meaning that if B ∈ Age(X ) and A ⊆ B isa substructure of B, then A ∈ Age(X ).


Age of a structure

It has “joint-embedding property”, meaning that ifA,B ∈ Age(X ), then there exists C ∈ Age(X ) containingboth A and B as substructures.

It has the “amalgamation property”, meaning that if we haveA,B,C ∈ Age(X ) such that A embeds as a substructure intoboth B and C , then there is D ∈ Age(X ) containing both Band C such that their common substructure A is identified inD.Formally, if ιB : A ↪→ B, ιC : A ↪→ C are the embeddings,then there are embeddings ρB : B ↪→ D and ρC : C ↪→ D suchthat ρC ◦ ιC = ρB ◦ ιB .


Fraısse theorem

Theorem (R. Fraısse, 1953)

Let C be a set of some finite structures. Suppose that C:

is countable (up to isomorphism),

is hereditary,

has the joint-embedding property,

has the amalgamation property.

Then there exists a countable homogeneous structure X , called theFraısse limit of C, such that Age(X ) = C.Moreover, for any homogeneous countable structure Y such thatAge(Y ) = C = Age(X ) we have that Y is isomorphic to X .


Examples

The following are examples of Fraısse classes:

the class of finite undirected graphs,

the class of finite partial orders, the class of finite distributivelattices, or the class of finite Boolean algebras,

the class of finite groups, or finitely presented groups.


Non-examples

The following are NOT examples of Fraısse classes:

the class of finitely generated groups,

the class of finite modular lattices,

the class of finite fields.


Metric homogeneous structures

Urysohn universal space - contains isometrically everyseparable metric space, finite partial isometries extend toautoisometries of the whole space- made from the class of finite rational metric spaces

Universal metric SIN group - contains isometrically as asubgroup every separable group with bounded invariantmetric, in particular it’s a universal Polish SIN group- made from the class of ‘finitely presented rational metricgroups’

Gurarij space- made from the class of finite-dimensional normed spaceswhose norm is a Minkowski functional of some polytope in Rn

with vertices having rational coordinates


Results

Theorem

There exists a ‘universal and homogeneous’ closed subspace of theGurarij space.Universality: There is a closed subspace H of the Gurarij space G,isometric to G, such that for every pair (E ,F ), where E is a closedsubspace of a Banach space F , there is a linear isometricembedding ι : F ↪→ G such that ι[E ] = ι[F ] ∩H.

- made from the same class of finite-dimensional spaces as in theplain Gurarij case, which are moreover equipped with 1-Lipschitzseminorms.


Results

Theorem

There exists a ‘universal and homogeneous’ 1-complementedsubspace of the Gurarij space together with a ‘universal andhomogeneous’ norm one projection.Universality: There is a norm one projection P on G onto a1-complemented H such that for every pair (E ,T ), where E is aseparable Banach space and T : E → E is a norm one projection,there is a linear isometric ι : E ↪→ G such that P ◦ ι = ι ◦ T .

- made from the same class of finite-dimensional spaces as in theplain Gurarij case, which are moreover equipped with 1-Lipschitzseminorms whose kernel is 1-complemented.


The universal operator on the Holmes’ space

Theorem

Let H be the Holmes’ space. Let Z be some fixed separableBanach space and L > 0 some constant. Then there exists auniversal operator φL on H to Z with norm L.

If X is a separable Banach space, ψ : X → Z is a linear operatorbounded by L, then there exists a linear isometric embeddingι : X → H such that φL ◦ ι = ψ.


The universal operator on the Holmes’ space

Theorem

Let H be the Holmes’ space. Let Z be some fixed separableBanach space and L > 0 some constant. Then there exists auniversal operator φL on H to Z with norm L.If X is a separable Banach space, ψ : X → Z is a linear operatorbounded by L, then there exists a linear isometric embeddingι : X → H such that φL ◦ ι = ψ.


Universal operator on the Holmes’ space

Fact

There exists a left-adjoint functor F to the forgetful functor fromthe category of Banach space (with bounded linear operators) tothe category of pointed metric spaces (with Lipschitz mapspreserving the distinguished point).

Let (M, 0) be a pointed metric space. There exists a Banach spaceF (M), containing M isometrically as a Hamel basis, with thefollowing property: let X be a Banach space and f : M → X aLipschitz map sending 0 to 0. Then f extends to a linear operatorf : F (M)→ X , with ‖f ‖ = Lip(f ), so that the following diagramcommutes:

M F (M)

X

δ

ff



Consider as M some separable Banach space Z , as X the freespace Z and as f the identity. Then we have:

Z F (Z )

Z

δ

idid = β

Theorem - Godefroy, Kalton

There exists a linear isometric embedding ι : Z → F (Z ) such thatβ ◦ ι = idZ .



Theorem

Let U be the Urysohn universal metric space. Let M be anyseparable metric space and L > 0 a positive constant. Then thereexists a universal Lipschitz map φL : U→ M with the Lipschitzconstant L.That is, for any separable metric space X and a Lipschitz mapf : X → M, with Lipf ≤ L, there exists an isometric embeddingi : M → U such that φL ◦ i = f , i.e. the following diagramcommutes:

X U

M

i

fφL



Theorem

Fix a separable Banach space Z and a constant L > 0. Letφ : U→ Z be the universal Lipschitz map of Lipschitz constant Lto Z . Then φ : F (U)→ Z is a universal operator of norm L on theHolmes’ space F (U).

Proof. Let X be a separable Banach space and f : X → Z a linearoperator of norm at most L. By the preceding theorem, thereexists an isometric embedding ι : X → U such that φ ◦ ι = f .Denote by X ′ ⊆ U the image ι[X ].By Godefroy, Kalton there is a linear isometric embeddingι : X → F (X ′) ⊆ F (U) (such that β ◦ ι = idX ).



We claim that ι : X → F (U) is the desired linear isometricembedding, i.e.

X F (U)

Z

ι

fφ

Indeed,

X F (X ′) ⊆ F (U)

Z

β

ιf

φ


Banach algebras problems

There seems to exist a left-adjoint functor F to the forgetfulfunctor from the category of Banach algebras to the category ofBanach spaces.That is, for any Banach space X there is a (unital) Banach algebraF (X ) and a linear embedding ι : X → F (X ) such that for anyBanach algebra A and a linear operator f : X → A there is aunique extension of f to a Banach algebra morphism f such thatthe following diagram commutes:

X F (X )

A

ι

ff



Set F (X ) =⊕

`1⊗kπX .

Has it been considered?



Do Banach algebras have amalgamation property? Is Fraıssethoery applicable?

Does there exist a universal separable Banach algebra?



Do Banach algebras have amalgamation property? Is Fraıssethoery applicable?Does there exist a universal separable Banach algebra?



Let X be a structure of some kind. A quantifier-free type over Xmay be viewed as a decription of some element x that lies in someextension X ⊆ X ′.

Examples: Let X be a metric space. A Katetov function on X is a1-Lipschitz function f : X → R+ satisfying dX (x , y) ≤ f (x) + f (y)for all x , y ∈ X . For any such function f we may define anextension X ∪ {xf } where we define the distances d(xf , y) as f (y),for y ∈ X .If G is a group with an invariant metric and f a Katetov functionon G , then there exists an extension of the invariant metric toG ∗ Z, where d(1Z, g) = f (g), where g ∈ G and 1Z is thegenerator of the copy of Z.If X is a Banach space and f is a convex Katetov function on X ,then there exists an extension of the norm to X ⊕ R where thedistance between the new vector and old elements of X isprescribed by f .



Let X be a structure of some kind. A quantifier-free type over Xmay be viewed as a decription of some element x that lies in someextension X ⊆ X ′.Examples: Let X be a metric space. A Katetov function on X is a1-Lipschitz function f : X → R+ satisfying dX (x , y) ≤ f (x) + f (y)for all x , y ∈ X . For any such function f we may define anextension X ∪ {xf } where we define the distances d(xf , y) as f (y),for y ∈ X .

If G is a group with an invariant metric and f a Katetov functionon G , then there exists an extension of the invariant metric toG ∗ Z, where d(1Z, g) = f (g), where g ∈ G and 1Z is thegenerator of the copy of Z.If X is a Banach space and f is a convex Katetov function on X ,then there exists an extension of the norm to X ⊕ R where thedistance between the new vector and old elements of X isprescribed by f .



Let X be a structure of some kind. A quantifier-free type over Xmay be viewed as a decription of some element x that lies in someextension X ⊆ X ′.Examples: Let X be a metric space. A Katetov function on X is a1-Lipschitz function f : X → R+ satisfying dX (x , y) ≤ f (x) + f (y)for all x , y ∈ X . For any such function f we may define anextension X ∪ {xf } where we define the distances d(xf , y) as f (y),for y ∈ X .If G is a group with an invariant metric and f a Katetov functionon G , then there exists an extension of the invariant metric toG ∗ Z, where d(1Z, g) = f (g), where g ∈ G and 1Z is thegenerator of the copy of Z.

If X is a Banach space and f is a convex Katetov function on X ,then there exists an extension of the norm to X ⊕ R where thedistance between the new vector and old elements of X isprescribed by f .



Let X be a structure of some kind. A quantifier-free type over Xmay be viewed as a decription of some element x that lies in someextension X ⊆ X ′.Examples: Let X be a metric space. A Katetov function on X is a1-Lipschitz function f : X → R+ satisfying dX (x , y) ≤ f (x) + f (y)for all x , y ∈ X . For any such function f we may define anextension X ∪ {xf } where we define the distances d(xf , y) as f (y),for y ∈ X .If G is a group with an invariant metric and f a Katetov functionon G , then there exists an extension of the invariant metric toG ∗ Z, where d(1Z, g) = f (g), where g ∈ G and 1Z is thegenerator of the copy of Z.If X is a Banach space and f is a convex Katetov function on X ,then there exists an extension of the norm to X ⊕ R where thedistance between the new vector and old elements of X isprescribed by f .



What are the quantifier-free types over Banach algebras? Are theyjust convex Katetov functions?(The problem seems to be very difficult for C ∗-algebras)


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