Transitivity and Ramsey properties of Lp

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Transitivity and Ramsey properties of L

_p

-spaces

Valentin Ferenczi Universidade de S ˜ao Paulo

Workshop on Banach spaces and Banach lattices ICMAT, September 9-13, 2019

Valentin Ferenczi Universidade de S ˜ao Paulo Lp spaces

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Outline

1. Transitivities of isometry groups

2. Fra¨ıss ´e theory and the KPT correspondence 3. Fra¨ıss ´e Banach spaces

4. The Approximate Ramsey Property for`ⁿ_p’s

Joint work with J. Lopez-Abad, B. Mbombo, S. Todorcevic Supported by Fapesp 2016/25574-8 and CNPq 30304/2015-7.

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Notation

X,Y denote an infinite dimensional Banach spaces,E,F,G,H finite dimensional ones.

S_X =unit sphere ofX.

GL(X)=the group of linear automorphisms ofX Isom(X)= group of linear surjective isometries ofX.

Emb(F,X)= set of linear isometric embeddings ofF intoX.

pa real number in the separable Banach range:1≤p<+∞

L_pdenotes the Lebesgue spaceL_p([0,1]).

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Outline

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Topologies on GL(X )

Three topologies are relevant onGL(X):

I the norm topologykTk=sup_x∈S

XkTxk.

I the SOT (strong operator topology) i.e. pointwise convergence onX,

I the WOT (weak operator topology) i.e. weak pointwise convergence onX,

Fact

GL(X),k.kis a topological group

It is not so forGL(X)with SOT in general, but things get better if one looks at bounded subgroups (in particularIsom(X)).

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The group Isom(X )

Fact

Isom(X)is a topological group for SOT.

ButIsom(X),k.kis not separable in general, forX separable.

Fact

If X is separable then(Isom(X),SOT)is separable. Actually it is a Polish group, i.e. SOT is separable completely metrizable

I Regarding WOT it coincides with SOT onIsom(X)as soon as X has PCP (e.g. reflexive) - see Megrelishvilii 2000 - so forL_p,1<p <∞.

I OnIsom(L₁)WOT and SOT are different (Antunes 2019)

I Things are more involved inC(K)-spaces, but those will not be studied here.

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The group Isom(X )

Fact

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The group Isom(X )

Fact

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The group Isom(X )

Summing up:

Isom(X)will always be equipped with theStrong Operator Topology SOT.

RegardingEmb(F,X), forF finite dimensional, we shall usually equipEmb(F,X)with the distance induced by thenorm on L(F,X). But note that here SOT and the norm topology are equivalent.

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Classical isometry groups

1 IfH=Hilbert, thenIsom(H)is the unitary groupU(H).

It actstransitivelyonS_H, meaning there is a single and full orbit for the actionIsom(H)yS_H.

2 For 1≤p<+∞,p6=2, every isometry onL_p=L_p(0,1)is of the form

T(f)(.) =h(.)f(φ(.)),

whereφis a measurable transformation of[0,1]onto itself, andhsuch that|h|^p =d(λ◦φ)/dλ,λthe Lebesgue measure (Banach-Lamperti 1932-1958). So

3 Isom(L_p)actsalmost transitivelyonS_L_p, meaning that the actionIsom(Lp)yS_L_p admitsdense orbits.

4 HoweverIsom(Lp)doesnotact almost transitively on the sphere unlessp=2.

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Classical isometry groups

T(f)(.) =h(.)f(φ(.)),

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Classical isometry groups

T(f)(.) =h(.)f(φ(.)),

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Classical isometry groups

T(f)(.) =h(.)f(φ(.)),

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Classical isometry groups

5 Every isometry onc₀and`p,p6=2 acts as a ”signed permutation”, i.e. a combination of signs and permutation of the coordinates on the canonical basis.

6 By the Banach-Stone theorem (1932), every isometry of C(K)is of the form

T(f)(.) =h(.)f(φ(.)),

wherehis continuous unimodular onK andφa homeomorphism ofK.

7 It follows thatIsom(`p)(resp. Isom(c₀), resp. Isom(C(K))) donotact almost transitively on the sphere. This somehow tells us that their isometry group is too rigid. Actually in the category ofL_p-spaces the relevant space isL_pforp<∞ and theGurarij spaceforp =∞.

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Classical isometry groups

T(f)(.) =h(.)f(φ(.)),

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Classical isometry groups

T(f)(.) =h(.)f(φ(.)),

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Mazur rotation problem

IfG=Isom(X)acts transitively onS_X, mustX be isomorphic?

isometric? to a Hilbert space.

(a) if dimX <+∞:YESto both

(b) if dimX = +∞and is separable: ???

(c) if dimX = +∞and is non-separable: NOto both

Proof

(a)Average a given inner product by using the Haar measure on G and observe that this new inner product turns all T ∈G into unitaries and therefore, by transitivity, must induce a multiple of the original norm.

[x,y] = Z

T∈G

<Tx,Ty >dµ(T),

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Mazur rotation problem

Proof

[x,y] = Z

T∈G

<Tx,Ty >dµ(T),

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Mazur rotation problem

Proof

[x,y] = Z

T∈G

<Tx,Ty >dµ(T),

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Mazur rotation problem

IfG=Isom(X)acts transitively onS_X, mustX be isometric?

isomorphic? to a Hilbert space.

Proof

(c)Use ultrapowers...

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Ultrapowers

A normed space istransitive (resp. almost transitive)if the associated isometry group acts transitively (resp. almost transitively) on the associated unit sphere.

It is an easy observation that ifX is almost transitive then for any non-principal ultrafilterU,X_U istransitive. Actually the subgroupIsom(X)U of isometriesT of the form

T((xn)n∈N) = (Tn(xn))n∈N

whereT_n∈Isom(X), acts transitively onXU. Proposition

The space(L_p(0,1))U is transitive.

Note that in these lines Cabello-Sanchez (1998) studies Πn∈NL_p_n(0,1)forp_n→+∞and obtains a transitive M-space.

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Ultrapowers

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Ultrapowers

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On renormings of classical spaces

Forp6=2,Lpis not transitive, and`pnot almost transitive.

Furthermore

Theorem (Dilworth - Randrianantoanina, 2014) Let1<p<+∞,p 6=2. Then

`_p does not admit an equivalent almost transitive norm.

Question

Let1≤p<+∞,p 6=2. Show that the space L_p([0,1])does not admit an equivalent transitive norm.

See also Cabello-Sanchez, Dantas, Kadets, Kim, Lee, Mart´ın (2019) for related notions of ”microtransitivity”.

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On renormings of classical spaces

Furthermore

Question

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On renormings of classical spaces

Furthermore

Question

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Ultrahomogeneity

Definition

LetX be a Banach space.

I X is calledultrahomogeneouswhen for every finite dimensional subspaceE ofX and every isometric embeddingφ:E →X there is a linear isometry

g ∈Isom(X)such thatg E =φ; this means the canonical actionIsom(X)yEmb(E,X)istransitive.

I X is called approximately ultrahomogeneous(AuH)when for every finite dimensional subspaceE ofX, every

isometric embeddingφ:E →X and everyε >0 there is a linear isometryg∈Isom(X)such thatkgE−φk< ε;

this means the canonical actionIsom(X)yEmb(E,X)is almost transitive(dense orbits).

The canonical action byg ∈Isom(X)onEmb(E,X)isφ7→g◦φ.

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Ultrahomogeneity

Definition

LetX be a Banach space.

I X is calledultrahomogeneouswhen for every finite dimensional subspaceE ofX and every isometric embeddingφ:E →X there is a linear isometry

g ∈Isom(X)such thatg E =φ; this means the canonical actionIsom(X)yEmb(E,X)istransitive.

I X is called approximately ultrahomogeneous(AuH)when for every finite dimensional subspaceE ofX, every

isometric embeddingφ:E →X and everyε >0 there is a linear isometryg∈Isom(X)such thatkgE−φk< ε;

this means the canonical actionIsom(X)yEmb(E,X)is almost transitive(dense orbits).

The canonical action byg ∈Isom(X)onEmb(E,X)isφ7→g◦φ.

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Examples

Note that ultrahomogeneous⇒transitive, and(AuH)⇒almost transitive

Fact

Any Hilbert space is ultrahomogeneous.

Theorem

Are(AuH), but not ultrahomogeneous:

I The Gurarij space, defined by Gurarij in 1966 (Kubis-Solecki 2013).

I Lp[0,1]for p6=2,4,6,8, . . . (Lusky 1978).

One original definition of the Gurarij:a separable Banach spaceGuniversal for f.d. spaces such that any linear isometry between f.d. subspaces extends to a1+-linear isometry on G. By Lusky 1976, it is isometrically unique.

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Examples

Note that ultrahomogeneous⇒transitive, and(AuH)⇒almost transitive

Fact

Any Hilbert space is ultrahomogeneous.

Theorem

Are(AuH), but not ultrahomogeneous:

I The Gurarij space, defined by Gurarij in 1966 (Kubis-Solecki 2013).

I Lp[0,1]for p6=2,4,6,8, . . . (Lusky 1978).

One original definition of the Gurarij:a separable Banach spaceGuniversal for f.d. spaces such that any linear isometry between f.d. subspaces extends to a1+-linear isometry on G. By Lusky 1976, it is isometrically unique.

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Examples

Theorem Are(AuH):

I The Gurarij space, defined by Gurarij in 1966 (Kubis-Solecki 2013)

I L_p[0,1]for p6=4,6,8, . . . (Lusky 1978) Note that

I the Gurarij is the unique separable, universal,(AuH)space (Lusky 1976 + Kubis-Solecki 2013).

I Lusky’s result abourL_p’s is based on theequimeasurability theoremby Plotkin / Rudin, 1976. His proof gives(AuH).

I Lp isnot(AuH)forp=4,6,8, . . .:

B. Randrianantoanina (1999) proved that for thosep⁰s there are two isometric subspaces ofL_p (due to

Rosenthal), with an unconditional basis, complemented/

uncomplemented.

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Examples

Theorem Are(AuH):

uncomplemented.

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Examples

Theorem Are(AuH):

uncomplemented.

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Examples

Theorem Are(AuH):

uncomplemented.

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A sketch of Lusky’s proof

It uses

Proposition (Plotkin and Rudin (1976))

For p∈/ 2N, suppose that(f₁, . . . ,f_n)∈L_p(Ω0,Σ0, µ0)and (g₁, . . . ,gn)∈Lp(Ω₁,Σ₁, µ₁)and

k1+

n

X

j=1

a_jf_jk_µ₀ =k1+

n

X

j=1

a_jg_jk_µ₁ for every a₁, . . . ,a_n.

Then(f₁, . . . ,f_n)and(g₁, . . . ,g_n)are equidistributed Equidistributed here means that for any BorelB∈Rⁿ,

µ0((f₁, . . . ,f_n)⁻¹(B)) =µ1((g₁, . . . ,g_n)⁻¹(B)).

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Lp’s for p non even are ”like” the Gurarij

Let us cite Lusky:

”We show that a certain homogeneity property holds for L_p(0,1);p6=4,6,8, . . ., which is similar to a corresponding property of the Gurarij space...”

We aim to give a more complete meaning to this similarity.

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Outline

2. Fra¨ıss ´e theory and the KPT correspondence

3. Fra¨ıss ´e Banach spaces

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Fra¨ıss ´e theory (abusively) summarized

I Given a (hereditary) classF of finite (or sometimes finitely generated) structures,Fra¨ıss ´e theory(Fra¨ıss ´e 1954) investigates the existence of a countable structureA, universal forF andultrahomogeneous(anytisomorphism between finite substructures extends to a global

automorphism ofA)

I Fra¨ıss ´e theory shows that this is equivalent to certain amalgamation properties ofF.

I ThenAisuniqueup to isomorphism and called theFra¨ıss ´e limitofF.

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Fra¨ıss ´e theory (abusively) summarized

automorphism ofA)

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Fra¨ıss ´e theory (abusively) summarized

automorphism ofA)

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Fra¨ıss ´e theory (abusively) summarized

Example

ifF=the class of finite sets, thenA=N

In this case isomorphisms of the structure are just bijections.

Example

ifF=the class of finite ordered sets, thenA= (Q, <).

Isomorphisms are order preserving bijections.

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Fra¨ıss ´e theory (abusively) summarized

Example

ifF=the class of finite sets, thenA=N

In this case isomorphisms of the structure are just bijections.

Example

ifF=the class of finite ordered sets, thenA= (Q, <).

Isomorphisms are order preserving bijections.

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Fra¨ıss ´e and Extreme Amenability

Fra¨ıss ´e theory is related toExtreme Amenabilitythrough the KPT correspondence(Kechris-Pestov-Todorcevic 2005).

Definition

A topological groupGis calledextremely amenable (EA)when every continuous actionGyK on a compactK has a fixed point; that is, there isp∈K such thatg·p=pfor allg ∈G.

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Examples of extremely amenable groups

1. The groupAut(Q, <)of strictly increasing bijections ofQ (with the pointwise convergence topology) (Pestov,1998);

2. butS∞=Aut(N)isnotextremely amenable;

3. The group of isometries of the Urysohn space with pointwise convergence topology. (Pestov, 2002);

4. The unitary groupU(H)endowed with SOT (Gromov-Milman,1983);

5. The groupIsom(L_p)of linear isometries of the Lebesgue spacesL_p[0,1], 1≤p 6=2<∞, with the SOT

(Giordano-Pestov, 2006);

6. The groupIsom(G)of linear isometries of the Gurarij space (Bartosova-LopezAbad-Lupini-Mbombo, 2017)

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Examples of extremely amenable groups

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Examples of extremely amenable groups

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Examples of extremely amenable groups

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Examples of extremely amenable groups

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Examples of extremely amenable groups

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The KPT correspondence

For finite structures, whenAis the Fra¨ıss ´e limit ofF, then holds theKechris-Pestov-Todorcevic correspondence.

Theorem (Kechris-Pestov-Todorcevic, 2005) AssumeF is”rigid”. Then the group

(Aut(A),ptwise cv topology ) is extremely amenable if and only ifF satisfies theRamsey property.

For example Pestov’s result thatAut(Q, <)is EA is a

combination of ”(Q, <) =Fra¨ıss ´e limit of finite ordered sets”

and of the classical finite Ramsey theorem onN.

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The KPT correspondence

For finite structures, whenAis the Fra¨ıss ´e limit ofF, then holds theKechris-Pestov-Todorcevic correspondence.

Theorem (Kechris-Pestov-Todorcevic, 2005) AssumeF is”rigid”. Then the group

(Aut(A),ptwise cv topology ) is extremely amenable if and only ifF satisfies theRamsey property.

For example Pestov’s result thatAut(Q, <)is EA is a

combination of ”(Q, <) =Fra¨ıss ´e limit of finite ordered sets”

and of the classical finite Ramsey theorem onN.

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Outline

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Fra¨ıss ´e Banach spaces

Several works exist about extension of the Fra¨ıss ´e theory to the metric setting (i.e. with epsilons), and settle the case of the Gurarij space,

(i.e. allow to see the Gurarij as the Fraiss ´e limit of the class of finite dimensional spaces)

but they are often at the same time too general and too

restrictive for us - and in particular do not apply in a satisfactory way to theL_p’s. We focus on the Banach space setting.

Anticipating here, note that there is no hope that classes of finite dimensional spaces are rigid, so only an Approximate Ramsey Property can be hoped for (think of concentration of measure).

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Fra¨ıss ´e Banach spaces

Several works exist about extension of the Fra¨ıss ´e theory to the metric setting (i.e. with epsilons), and settle the case of the Gurarij space,

(i.e. allow to see the Gurarij as the Fraiss ´e limit of the class of finite dimensional spaces)

but they are often at the same time too general and too

restrictive for us - and in particular do not apply in a satisfactory way to theL_p’s. We focus on the Banach space setting.

Anticipating here, note that there is no hope that classes of finite dimensional spaces are rigid, so only an Approximate Ramsey Property can be hoped for (think of concentration of measure).

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Fra¨ıss ´e Banach spaces

Given two Banach spacesE andX, andδ ≥0, letEmbδ(E,X) be the collection of all linearδ-isometricembeddings

T :E →X, i.e. such thatkTk,kT⁻¹k ≤1+δ (T⁻¹defined on T(E)), equipped with the distance induced by the norm.

We consider the canonical actionIsom(X)yEmb_δ(E,X)

Definition (F., Lopez-Abad, Mbombo, Todorcevic)

X isFra¨ıss ´eif and only if for everyk ∈Nand everyε >0 there isδ >0 such that for everyE ⊂X of dimensionk, the action Isom(X)yEmbδ(E,X)is ”ε-transitive”

(i.e. everyδ-isometric embedding ofE intoX is in the ε-expansion of any given orbit).

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Fra¨ıss ´e Banach spaces

Given two Banach spacesE andX, andδ ≥0, letEmbδ(E,X) be the collection of all linearδ-isometricembeddings

T :E →X, i.e. such thatkTk,kT⁻¹k ≤1+δ (T⁻¹defined on T(E)), equipped with the distance induced by the norm.

We consider the canonical actionIsom(X)yEmb_δ(E,X)

Definition (F., Lopez-Abad, Mbombo, Todorcevic)

(i.e. everyδ-isometric embedding ofE intoX is in the ε-expansion of any given orbit).

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Fra¨ıss ´e Banach spaces

Definition

Note that Fra¨ıss ´e⇒(AuH) Proposition

TFAE for X :

I X is Fra¨ıss ´e

I X is ”weak Fra¨ıss ´e”, i.e. as in the Fra¨ıss ´e definition, but assuming thatδdepends onεand E (instead ofdimE ), and

eachAge_k(X)is compact in the Banach-Mazur pseudo-distance.

Age_k(X) =set ofk-dim. subspaces ofX.

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Fra¨ıss ´e Banach spaces

Definition

Note that Fra¨ıss ´e⇒(AuH) Proposition

TFAE for X :

I X is ”weak Fra¨ıss ´e”, i.e. as in the Fra¨ıss ´e definition, but assuming thatδdepends onεand E (instead ofdimE ), and

eachAge_k(X)is compact in the Banach-Mazur pseudo-distance.

Age_k(X) =set ofk-dim. subspaces ofX.

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Examples of Fra¨ıss ´e spaces

I Hilbert spaces are Fra¨ıss ´e (ε=δ, exercise);

I the Gurarij space is Fra¨ıss ´e (actuallyε=2δ) ;

I L_p isnotFra¨ıss ´e forp=4,6,8, . . . since not AUH.

Sinceεdepends only onδand not onn, we say that the Hilbert and the Gurarij are ”stable” Fra¨ıss ´e”.

On the other hand, Theorem

(F.,Lopez-Abad, Mbombo, Todorcevic) The spaces L_p[0,1]for p6=4,6,8, . . . are Fra¨ıss ´e.

How can we get convinced that this is the relevant definition?

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Examples of Fra¨ıss ´e spaces

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Examples of Fra¨ıss ´e spaces

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Examples of Fra¨ıss ´e spaces

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Examples of Fra¨ıss ´e spaces

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Properties of Fra¨ıss ´e spaces

Proposition

Assume Y is Fraiss ´e, and that X is separable. Then are equivalent:

(1) X is finitely representable in Y

(2) every finite dimensional subspace of X embeds isometrically into Y

(3) X embeds isometrically in Y

In particular (by Dvoretsky)`₂is the minimal separable Fra¨ıss ´e space; and the Gurarij is the maximal one.

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Properties of Fra¨ıss ´e spaces

Proposition

Assume Y is Fraiss ´e, and that X is separable. Then are equivalent:

(1) X is finitely representable in Y

(2) every finite dimensional subspace of X embeds isometrically into Y

(3) X embeds isometrically in Y

In particular (by Dvoretsky)`₂is the minimal separable Fra¨ıss ´e space; and the Gurarij is the maximal one.

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Properties of Fra¨ıss ´e spaces

Let

I Age(X)=the set of finite dimensional subspaces ofX, and

I forF,Gclasses of finite dimensional spaces,F ≡ G mean that any element ofF has an isometric copy inG and conversely.

Proposition

Assume X and Y are separable Fra¨ıss ´e. Then are equivalent (1) X is finitely representable in Y and vice-versa,

(2) Age(X)≡Age(Y), (3) X and Y are isometric.

So separable Fraiss ´e spaces areuniquely determinedby their age modulo≡.

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Properties of Fra¨ıss ´e spaces

Let

Proposition

So separable Fraiss ´e spaces areuniquely determinedby their age modulo≡.

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Properties of Fra¨ıss ´e spaces

Let

Proposition

We also obtained internal characterizations of classes of finite dimensional spaces which are≡to the age of some Fra¨ıss ´e (”amalgamation properties”). For such a classF we write X=Fra¨ıss ´e limF to mean ”X separable andAge(X)≡ F”

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Fra¨ıss ´e is an ultraproperty

Proposition

The following are equivalent.

1) X is weak Fra¨ıss ´e.

2) For every E ∈Age(X)the action(Isom(X))U yEmb(E,XU) is (almost) transitive.

Furthermore, the following are equivalent:

1) X is Fra¨ıss ´e.

2) For every E ∈Age(XU)the action(Isom(X))U yEmb(E,XU) is (almost) transitive.

3) For every separable Z ⊂XU the action (Isom(X))U yEmb(Z,X_U)is transitive.

4) XU is Fra¨ıss ´e and(Isom(X))U is SOT-dense inIsom(XU)

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Fra¨ıss ´e is an ultraproperty

Proposition

The following are equivalent.

1) X is weak Fra¨ıss ´e.

2) For every E ∈Age(X)the action(Isom(X))U yEmb(E,XU) is (almost) transitive.

Furthermore, the following are equivalent:

1) X is Fra¨ıss ´e.

2) For every E ∈Age(XU)the action(Isom(X))U yEmb(E,XU) is (almost) transitive.

3) For every separable Z ⊂XU the action (Isom(X))U yEmb(Z,X_U)is transitive.

4) XU is Fra¨ıss ´e and(Isom(X))U is SOT-dense inIsom(XU)

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Fra¨ıss ´e is an ultraproperty

In particular, it follows that ifX is Fra¨ıss ´e, then its ultrapowers are Fra¨ıss ´e and ultrahomogeneous.

Corollary

The non-separable L_p-space(L_p(0,1))_U is ultrahomogeneous.

A similar fact was observed for the Gurarij, by Aviles, Cabello, Castillo, Gonzalez, Moreno, 2013.

This is related to the theory of ”strong Gurarij” spaces (Kubis).

Note: they must be non-separable.

Question

Is there a non-Hilbertian separable ultrahomogeneous space?

an ultrahomogeneous renorming of Lp(0,1)?

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Fra¨ıss ´e is an ultraproperty

Corollary

Question

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Fra¨ıss ´e is an ultraproperty

Corollary

Question

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Lp spaces are Fra¨ıss ´e, p 6= 4, 6, 8, . . .

Note the result by G. Schechtman 1979 (+ Dor 1975 forp=1) - as observed by D. Alspach 1983.

Theorem (Dor - Schechtman)

For any1≤p<∞anyε >0, there existsδ=δp()>0such that

Emb_δ(`ⁿ_p,Lp(µ))⊂(Emb(`ⁿ_p,Lp(µ)))_ε. for every n∈N, and finite measureµ.

So the Fra¨ıss ´e property inL_pis satisfied in a strong sense for subspaces isometric to an`ⁿ_p.

Note however that Schechtman’s result holds forp =4,6,8, . . ., so things have to be more complicated for other subspaces and p6=4,6,8, . . ..

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Lp spaces are Fra¨ıss ´e, p 6= 4, 6, 8, . . .

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Lp spaces are Fra¨ıss ´e, p 6= 4, 6, 8, . . .

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Lp spaces are Fra¨ıss ´e, p 6= 4, 6, 8, . . .

Recall:

Proposition TFAE for X :

I Isom(X)yEmb_δ(E,X)isε-transitive for someδdepending onεand E

eachAge_k(X)is compact in the Banach-Mazur pseudodistance, whereAge_k(X) =class of k -dim.

subspaces of X .

I It is known thatAge_k(Lp)is closed in Banach-Mazur.

(actuallyAge_k(X)is closed⇔Age_k(X) =Age_k(XU)).

I So we only need to show thatIsom(X)yEmb_δ(E,X)is

ε-transitive for someδ depending onεandE.

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Lp spaces are Fra¨ıss ´e, p 6= 4, 6, 8, . . .

Recall:

Proposition TFAE for X :

I Isom(X)yEmb_δ(E,X)isε-transitive for someδdepending onεand E

eachAge_k(X)is compact in the Banach-Mazur pseudodistance, whereAge_k(X) =class of k -dim.

subspaces of X .

I It is known thatAge_k(Lp)is closed in Banach-Mazur.

(actuallyAge_k(X)is closed⇔Age_k(X) =Age_k(XU)).

I So we only need to show thatIsom(X)yEmb_δ(E,X)is ε-transitive for someδ depending onεandE.

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Lp spaces are Fra¨ıss ´e, p 6= 4, 6, 8, . . .

Recall

For p∈/ 2N, suppose that(f₁, . . . ,fn)∈Lp(Ω₀,Σ₀, µ₀)and (g₁, . . . ,g_n)∈L_p(Ω₁,Σ₁, µ₁)and

k1+

n

X

j=1

a_jf_jk_µ₀ =k1+

n

X

j=1

Then(f₁, . . . ,f_n)and(g₁, . . . ,g_n)are equidistributed

(i.e.µ0((f1(ω), . . . ,fn(ω))∈B) =µ1((g1(ω), . . . ,gn(ω))∈B)for every B⊂RⁿBorel)

Also recall that we sketched the proof by Lusky that Corollary

Those L_p’s are (AuH).

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Lp spaces are Fra¨ıss ´e, p 6= 4, 6, 8, . . .

Recall

For p∈/ 2N, suppose that(f₁, . . . ,fn)∈Lp(Ω₀,Σ₀, µ₀)and (g₁, . . . ,g_n)∈L_p(Ω₁,Σ₁, µ₁)and

k1+

n

X

j=1

a_jf_jk_µ₀ =k1+

n

X

j=1

Then(f₁, . . . ,f_n)and(g₁, . . . ,g_n)are equidistributed

(i.e.µ0((f1(ω), . . . ,fn(ω))∈B) =µ1((g1(ω), . . . ,gn(ω))∈B)for every B⊂RⁿBorel)

Also recall that we sketched the proof by Lusky that Corollary

Those L_p’s are (AuH).

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Lp spaces are Fra¨ıss ´e, p 6= 4, 6, 8, . . .

To prove that thoseLp’s are Fra¨ıss ´e, the main step is to prove a

”continuous” version of Plotkin-Rudin, in the sense that if (1+δ)⁻¹k1+

n

X

j=1

a_jg_jk_µ₁ ≤ k1+

n

X

j=1

a_jf_jk_µ₀ ≤(1+δ)k1+

n

X

j=1

a_jg_jk_µ₁

then(f₁, . . . ,fn)and(g₁, . . . ,gn)are ”ε-equimeasurable” in some sense.

more precisely, we measure proximity of associated measures onRⁿin the L ´evy-Prokhorov metric.

dLP(µ, ν) :=inf{ε >0|µ(A)≤ν(A_ε) +εandν(A)≤µ(A_ε) +ε∀A}.

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Fraiss ´e limits of non hereditary classes

It is also possible and useful to develop a Fra¨ıss ´e theory with respect to certain classes of finite dimensional subspaces, which are not≡to the Age of anyX, because they are not hereditary.

ForL_p(0,1)we can use the family of`ⁿ_p’s and the perturbation result of Dor -Schechtmann to give meaning to

Theorem

Forany1≤p<+∞,

Lp=lim`ⁿ_p.

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Fraiss ´e limits of non hereditary classes

It is also possible and useful to develop a Fra¨ıss ´e theory with respect to certain classes of finite dimensional subspaces, which are not≡to the Age of anyX, because they are not hereditary.

ForL_p(0,1)we can use the family of`ⁿ_p’s and the perturbation result of Dor -Schechtmann to give meaning to

Theorem

Forany1≤p<+∞,

Lp=lim`ⁿ_p.

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Fra¨ıss ´e Banach lattices

By consideringlattice embeddingsand appropriate notions of δ-lattice embeddings, we may develop a Fra¨ıss ´e theory in the lattice setting, definingFra¨ıss ´e Banach lattices, i.e. some unique universal object for classes of finite dimensional lattices with an approximate lattice ultrahomogeneity property.

For example for 1≤p<+∞,Lp(0,1)is a Fra¨ıss ´e Banach lattice.

This means exactly that it is the Fra¨ıss ´e lattice limit of its finite sublattices the`ⁿ_p’s.

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Fra¨ıss ´e Banach lattices

By consideringlattice embeddingsand appropriate notions of δ-lattice embeddings, we may develop a Fra¨ıss ´e theory in the lattice setting, definingFra¨ıss ´e Banach lattices, i.e. some unique universal object for classes of finite dimensional lattices with an approximate lattice ultrahomogeneity property.

For example for 1≤p<+∞,Lp(0,1)is a Fra¨ıss ´e Banach lattice.

This means exactly that it is the Fra¨ıss ´e lattice limit of its finite sublattices the`ⁿ_p’s.

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A related construction: the lattice Gurarij

Recall that theGurarij spaceis obtained as the Fraiss ´e limit of the class of finite dimensional normed spaces, or equivalently, as the limit of the class of spaces isometric to`ⁿ_∞’s. See Bartosova - Lopez-Abad - Mbombo - Todorcevic (2017).

The point here is that isometric embeddings between`ⁿ_p’s respect the lattice structure ifp <+∞, butnotifp= +∞.

As Fra¨ıss ´e limit of the`ⁿ_∞’s with isometric lattice embeddings we obtain a new object that we call thelattice Gurarij.

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A related construction: the lattice Gurarij

Recall that theGurarij spaceis obtained as the Fraiss ´e limit of the class of finite dimensional normed spaces, or equivalently, as the limit of the class of spaces isometric to`ⁿ_∞’s. See Bartosova - Lopez-Abad - Mbombo - Todorcevic (2017).

The point here is that isometric embeddings between`ⁿ_p’s respect the lattice structure ifp <+∞, butnotifp= +∞.

As Fra¨ıss ´e limit of the`ⁿ_∞’s with isometric lattice embeddings we obtain a new object that we call thelattice Gurarij.