Classical summable ideals

Motivation Polish- and Banach-representability Representability inc₀and`₁

Representations of ideals in Banach spaces

Barnabás Farkas¹

2joint work with

3Piotr Borodulin-Nadzieja² and Grzegorz Plebanek²

4First Brazilian Workshop in Geometry of Banach Spaces

1Kurt Gödel Research Center, University of Vienna

2Mathematical Institute, University of Wrocław

Motivation Polish- and Banach-representability Representability inc₀and`₁

Classical summable ideals

Definition

Leth:ω→[0,∞)be a sequence such thatP

n∈ωh(n) =∞.

Then thesummable ideal associated tohis I_h=

A⊆ω:X

n∈A

h(n)<∞

(anF_σ P-ideal).

We have to change the definition:A∈ I_h⁰ if the sum of(h(n))n∈A

isunconditionally convergent, that is, the net XhA=

s_h(F) =X

n∈F

h(n) :F ∈[A]^<ω

is convergent.

But we still have the same family of ideals:A∈ I_h⁰ iffA∈ I_|h|. . .

Motivation Polish- and Banach-representability Representability inc₀and`₁

Classical summable ideals

Definition

Leth:ω→[0,∞)be a sequence such thatP

n∈ωh(n) =∞.

Then thesummable ideal associated tohis I_h=

A⊆ω:X

n∈A

h(n)<∞

(anF_σ P-ideal).

Whyh:ω →[0,∞)? What ifh:ω→R?

We have to change the definition:A∈ I_h⁰ if the sum of(h(n))n∈A

isunconditionally convergent, that is, the net XhA=

s_h(F) =X

n∈F

h(n) :F ∈[A]^<ω

is convergent.

But we still have the same family of ideals:A∈ I_h⁰ iffA∈ I_|h|. . .

Classical summable ideals

Definition

Leth:ω→[0,∞)be a sequence such thatP

n∈ωh(n) =∞.

Then thesummable ideal associated tohis I_h=

A⊆ω:X

n∈A

h(n)<∞

(anF_σ P-ideal).

Whyh:ω →[0,∞)? What ifh:ω→R?

We have to change the definition:A∈ I_h⁰ if the sum of(h(n))n∈A

isunconditionally convergent, that is, the net XhA=

s_h(F) =X

n∈F

h(n) :F ∈[A]^<ω

is convergent.

But we still have the same family of ideals:A∈ I_h⁰ iffA∈ I_|h|. . .

Motivation Polish- and Banach-representability Representability inc₀and`₁

Generalized summable ideals

h : ω → ?

Definition

LetGbe a Polish Abelian group andh:ω→Gsuch thatP his not convergent. Then thegeneralized summable ideal

associated toGandhis I_h^G =n

A⊆ω:X

hA is convergento .

We say that an idealJ onωisrepresentable inGif there is anh:ω →Gsuch thatJ =I_h^G. IfCis a class of groups then J is C-representableif it is representable in aG∈C.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Generalized summable ideals

h : ω → ?

Definition

LetGbe a Polish Abelian group andh:ω→Gsuch thatP his not convergent. Then thegeneralized summable ideal

associated toGandhis I_h^G =n

A⊆ω:X

hA is convergento .

anh:ω →Gsuch thatJ =I_h^G. IfCis a class of groups then J is C-representableif it is representable in aG∈C.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Generalized summable ideals

h : ω → ?

Definition

LetGbe a Polish Abelian group andh:ω→Gsuch thatP his not convergent. Then thegeneralized summable ideal

associated toGandhis I_h^G =n

A⊆ω:X

hA is convergento .

We say that an idealJ onωisrepresentable inGif there is anh:ω →Gsuch thatJ =I_h^G. IfCis a class of groups then J is C-representableif it is representable in aG∈C.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Examples

Example

J is representable inRⁿ(or inTⁿ) iffJ is a summable ideal.

J is representable inR^ω (or inT^ω) iffJ is the intersection of countable many summable ideals. These ideals are not necessarilyF_σ.

Example

If(G_k)_k∈ω is a sequence of non-trivial discrete Abelian groups, thenJ is representable inQ

k∈ωG_k iffJ is representable inZ^ω₂ iff there is a family{X_k:k ∈ω} ⊆[ω]^ω such that

J = n

A⊆ω:∀k ∈ω|A∩X_k|< ω o

.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Examples

Example

J is representable inRⁿ(or inTⁿ) iffJ is a summable ideal.

Example

J is representable inR^ω (or inT^ω) iffJ is the intersection of countable many summable ideals. These ideals are not necessarilyF_σ.

Example

If(G_k)_k∈ω is a sequence of non-trivial discrete Abelian groups, thenJ is representable inQ

k∈ωG_k iffJ is representable inZ^ω₂ iff there is a family{X_k:k ∈ω} ⊆[ω]^ω such that

J = n

A⊆ω:∀k ∈ω|A∩X_k|< ω o

.

Examples

Example

J is representable inRⁿ(or inTⁿ) iffJ is a summable ideal.

Example

J is representable inR^ω (or inT^ω) iffJ is the intersection of countable many summable ideals. These ideals are not necessarilyF_σ.

Example

If(G_k)_k∈ω is a sequence of non-trivial discrete Abelian groups, thenJ is representable inQ

k∈ωG_k iffJ is representable inZ^ω₂ iff there is a family{X_k:k ∈ω} ⊆[ω]^ω such that

J = n

A⊆ω:∀k ∈ω|A∩X_k|< ω o

.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Examples

Example

Thedensity zero ideal Z =

A⊆ω: |A∩n|

n →0

=

A⊆ω: |A∩[2ⁿ,2ⁿ⁺¹)|

2ⁿ →0

(anF_σδ P-ideal) is representable inc₀:

Examples

h(0) =(0,0, 0 , 0 , 0 , . . .) h(1) =(0,1, 0 , 0 , 0 , . . .) h(2) =(0,0,1/2, 0 , 0 , . . .) h(3) =(0,0,1/2, 0 , 0 , . . .) h(4) =(0,0, 0 ,1/4, 0 , . . .) h(5) =(0,0, 0 ,1/4, 0 , . . .) h(6) =(0,0, 0 ,1/4, 0 , . . .) h(7) =(0,0, 0 ,1/4, 0 , . . .)

... ... ... ... ... ... IfA⊆ω thenP

hAis convergent ⇐⇒

X

n∈A

h(n) =

0,|A∩[2,4)|

2 ,|A∩[4,8)|

4 , . . .

∈c₀ ⇐⇒ A∈ Z.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Why are we doing this?

(i) Our approach reveals some “geometric” properties of ideals and therefore it can be helpful in classifying certain classes of ideals.

(ii) These methods can be useful in providing new interesting examples of non-pathological analytic P-ideals (see later).

(iii) Representability of certain ideals in a Banach space can be seen as a combinatorial property of the space itself and this may lead us to develop new methods in the theory of Banach spaces.

Polish- and Banach-representability

Theorem

J is Polish-representable iffJ is an analytic P-ideal.

Theorem

J is Banach-representable iffJ is a non-path. analytic P-ideal.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Analytic P-ideals

A functionϕ:P(ω)→[0,∞]is

asubmeasure(onω) ifϕ(∅) =0,ϕis monotonic, subadditive, andϕ({n})<∞for everyn∈ω;

lower semicontinuous(lsc, in short) if ϕ(X) =limn→∞ϕ(X ∩n)for eachX ⊆ω.

Ifϕis an lsc submeasure then let Fin(ϕ)=

n

A⊆ω:ϕ(A)<∞o

(anF_σ ideal).

Exh(ϕ)= n

A⊆ω: lim

n→∞ϕ(A\n) =0 o

(anF_σδ P-ideal).

Analytic P-ideals

Theorem (Mazur, Solecki)

LetJ be an ideal onω. Then the following are equivalent:

(i) J is an analyticP-ideal.

(ii) J =Exh(ϕ)for some (finite) lsc submeasureϕ.

(iii) There is a Polish group topology onJ (with respect to4)

such that the Borel structure of this topology coincides with the Borel structure inherited fromP(ω).

Furthermore,J is anFσ P-ideal iffJ =Fin(ϕ) =Exh(ϕ)for some lsc submeasureϕ.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Examples

Classical summable ideals.

(Generalized) Density ideals: Let~µ= (µ_n)n∈ω be a sequence of (sub)measures onω with pairwise disjoint finite supports with lim sup_n→∞µn(ω)>0. Then the (generalized) density ideal associated to~µis

Z_~µ=

A⊆ω:µn(A)→0 . The trace of the null ideal:

tr(N)=

A⊆2^<ω:λ{f ∈2^ω:∃^∞n f n∈A}=0 . Farah’s ideal:

J_F =

A⊆ω:X

n∈ω

min{n,|A∩[2ⁿ,2ⁿ⁺¹)|} n² <∞

.

Tsirelson ideals.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Examples

Classical summable ideals.

(Generalized) Density ideals: Let~µ= (µ_n)n∈ω be a sequence of (sub)measures onω with pairwise disjoint finite supports with lim sup_n→∞µn(ω)>0. Then the (generalized) density ideal associated to~µis

Z_~µ=

A⊆ω:µn(A)→0 .

tr(N)=

A⊆2^<ω:λ{f ∈2^ω:∃^∞n f n∈A}=0 . Farah’s ideal:

J_F =

A⊆ω:X

n∈ω

min{n,|A∩[2ⁿ,2ⁿ⁺¹)|} n² <∞

.

Tsirelson ideals.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Examples

Classical summable ideals.

(Generalized) Density ideals: Let~µ= (µ_n)n∈ω be a sequence of (sub)measures onω with pairwise disjoint finite supports with lim sup_n→∞µn(ω)>0. Then the (generalized) density ideal associated to~µis

Z_~µ=

A⊆ω:µn(A)→0 . The trace of the null ideal:

tr(N)=

A⊆2^<ω:λ{f ∈2^ω:∃^∞n f n∈A}=0 .

Farah’s ideal: J_F =

A⊆ω:X

n∈ω

min{n,|A∩[2ⁿ,2ⁿ⁺¹)|} n² <∞

.

Tsirelson ideals.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Examples

Classical summable ideals.

(Generalized) Density ideals: Let~µ= (µ_n)n∈ω be a sequence of (sub)measures onω with pairwise disjoint finite supports with lim sup_n→∞µn(ω)>0. Then the (generalized) density ideal associated to~µis

Z_~µ=

A⊆ω:µn(A)→0 . The trace of the null ideal:

tr(N)=

A⊆2^<ω:λ{f ∈2^ω:∃^∞n f n∈A}=0 . Farah’s ideal:

J_F =

A⊆ω:X

n∈ω

min{n,|A∩[2ⁿ,2ⁿ⁺¹)|}

n² <∞

.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Examples

Classical summable ideals.

(Generalized) Density ideals: Let~µ= (µ_n)n∈ω be a sequence of (sub)measures onω with pairwise disjoint finite supports with lim sup_n→∞µn(ω)>0. Then the (generalized) density ideal associated to~µis

Z_~µ=

A⊆ω:µn(A)→0 . The trace of the null ideal:

tr(N)=

A⊆2^<ω:λ{f ∈2^ω:∃^∞n f n∈A}=0 . Farah’s ideal:

J_F =

A⊆ω:X

n∈ω

min{n,|A∩[2ⁿ,2ⁿ⁺¹)|}

n² <∞

.

Tsirelson ideals.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Non-pathological ideals

A submeasureϕis non-pathologicalif for everyA⊆ω ϕ(A) =sup

µ(A) :µis a measure on P(ω) and µ≤ϕ . An analytic P-idealJ isnon-pathologicalifJ =Exh(ϕ)for some non-pathological lsc submeasureϕ.

An analytic P-idealJ is non-pathological iffJ X ≤_K Z(i.e. there is anf :ω→X such thatf⁻¹[A]∈ Z for everyA∈ J) for everyX ∈ P(ω)\ J.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Non-pathological ideals

A submeasureϕis non-pathologicalif for everyA⊆ω ϕ(A) =sup

µ(A) :µis a measure on P(ω) and µ≤ϕ . An analytic P-idealJ isnon-pathologicalifJ =Exh(ϕ)for some non-pathological lsc submeasureϕ.

Theorem (Hrušák)

An analytic P-idealJ is non-pathological iffJ X ≤_K Z(i.e.

there is anf :ω→X such thatf⁻¹[A]∈ Z for everyA∈ J) for everyX ∈ P(ω)\ J.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Representability in c

Proposition

An idealJ is representable inc₀iff there is a family

{µ_k :k ∈ω}of measures onωsuch thatJ =Exh(ϕ)where ϕ=sup{µ_k :k ∈ω}and{k :n∈supp(µ_k)}is finite for eachn.

Density ideals are representable inc₀.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Representability in c

Proposition

An idealJ is representable inc₀iff there is a family

{µ_k :k ∈ω}of measures onωsuch thatJ =Exh(ϕ)where ϕ=sup{µ_k :k ∈ω}and{k :n∈supp(µ_k)}is finite for eachn.

Example

Density ideals are representable inc₀.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Representability in c

We show thattr(N), Farah’s ideal, and Tsirelson ideals are not representable inc₀.

IfJ is representable inc₀andtotally bounded(that is, if J =Exh(ϕ)thenϕ(ω)<∞), thenJ is a generalized density ideal.

Corollary

tr(N)is not representable inc₀. Proof:

1.tr(N)is not a generalized density ideal (because e.g. it is summable-like).

2. Hrusak+Hernandez-Hernandez:tr(N)is totally bounded (because e.g.s(tr(N)) =ω).

Motivation Polish- and Banach-representability Representability inc₀and`₁

Representability in c

We show thattr(N), Farah’s ideal, and Tsirelson ideals are not representable inc₀.

Proposition

IfJ is representable inc₀andtotally bounded(that is, if J =Exh(ϕ)thenϕ(ω)<∞), thenJ is a generalized density ideal.

Corollary

tr(N)is not representable inc₀. Proof:

1.tr(N)is not a generalized density ideal (because e.g. it is summable-like).

2. Hrusak+Hernandez-Hernandez:tr(N)is totally bounded (because e.g.s(tr(N)) =ω).

Representability in c

We show thattr(N), Farah’s ideal, and Tsirelson ideals are not representable inc₀.

Proposition

IfJ is representable inc₀andtotally bounded(that is, if J =Exh(ϕ)thenϕ(ω)<∞), thenJ is a generalized density ideal.

Corollary

tr(N)is not representable inc₀. Proof:

1.tr(N)is not a generalized density ideal (because e.g. it is summable-like).

2. Hrusak+Hernandez-Hernandez:tr(N)is totally bounded (because e.g.s(tr(N)) =ω).

Motivation Polish- and Banach-representability Representability inc₀and`₁

Representability in c

Theorem

If a tallFσ ideal is representable inc₀iff it is a summable ideal.

Corollary

Farah’s ideal and Tsirelson ideals are not representable inc₀.

Representability in c

Theorem

If a tallFσ ideal is representable inc₀iff it is a summable ideal.

Corollary

Farah’s ideal and Tsirelson ideals are not representable inc₀.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Representability in `

Theorem

There is a tallF_σ P-ideal (theRademacher-ideal) which is representable in`₁but not a summable ideal.

Proof (sketch): Our first, still unfinished (but 99% working,) idea was to build blocks of the form

(0, . . . ,0, ε₀/an, . . . , εn−1/an,0, . . .)whereε_i ∈ {±1}andanis an appropriate sequence tending to∞. Problem: If we add all 2ⁿpossible new elements to our family, then we should understand the following constant:

infn_max{kP

Fk:∅6=F⊆S}

P{kvk:v∈S} :∅ 6=S∈[X\ {0}]^<ωo .

Piotr’s refinement: At thenth stage we add “Rademacher-like” vectors only and then we can apply Khintchine’s inequality etc.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Representability in `

Theorem

There is a tallF_σ P-ideal (theRademacher-ideal) which is representable in`₁but not a summable ideal.

idea was to build blocks of the form

(0, . . . ,0, ε₀/an, . . . , εn−1/an,0, . . .)whereε_i ∈ {±1}andanis an appropriate sequence tending to∞. Problem: If we add all 2ⁿpossible new elements to our family, then we should understand the following constant:

infn_max{kP

Fk:∅6=F⊆S}

P{kvk:v∈S} :∅ 6=S∈[X\ {0}]^<ωo .

Piotr’s refinement: At thenth stage we add “Rademacher-like” vectors only and then we can apply Khintchine’s inequality etc.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Representability in `

Theorem

There is a tallF_σ P-ideal (theRademacher-ideal) which is representable in`₁but not a summable ideal.

Proof (sketch): Our first, still unfinished (but 99% working,) idea was to build blocks of the form

(0, . . . ,0, ε₀/an, . . . , εn−1/an,0, . . .)whereε_i ∈ {±1}andanis an appropriate sequence tending to∞. Problem: If we add all 2ⁿpossible new elements to our family, then we should understand the following constant:

infn_max{kP

Fk:∅6=F⊆S}

P{kvk:v∈S} :∅ 6=S∈[X\ {0}]^<ωo .

Piotr’s refinement: At thenth stage we add “Rademacher-like” vectors only and then we can apply Khintchine’s inequality etc.

Representability in `

Theorem

There is a tallF_σ P-ideal (theRademacher-ideal) which is representable in`₁but not a summable ideal.

Proof (sketch): Our first, still unfinished (but 99% working,) idea was to build blocks of the form

(0, . . . ,0, ε₀/an, . . . , εn−1/an,0, . . .)whereε_i ∈ {±1}andanis an appropriate sequence tending to∞. Problem: If we add all 2ⁿpossible new elements to our family, then we should understand the following constant:

infn_max{kP

Fk:∅6=F⊆S}

P{kvk:v∈S} :∅ 6=S∈[X\ {0}]^<ωo .

Piotr’s refinement: At thenth stage we add “Rademacher-like”

vectors only and then we can apply Khintchine’s inequality etc.

Motivation Polish- and Banach-representability Representability inc₀and`₁

Questions

Question

General version: Is there any reasonable characterization of R^ω-, orc₀-, or`1-representable ideals?

Reasonable versions:

Which ideals can be covered by a summable ideal?

Are all`1-representable idealsF_σ? Question

Assume that all non-pathological analytic P-ideals are representable in a Banach spaceX. Does it imply thatX is universal for the class of all separable Banach spaces?

Question

What can we say about ideals representable in (completions of) weak topologies?

Thank you for your attention!