Geometry of analytic P-ideals Piotr Borodulin-Nadzieja Vienna 2013 - - PowerPoint PPT Presentation

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Geometry of analytic P-ideals

Piotr Borodulin-Nadzieja

Vienna 2013

joint work with Barnab´ as Farkas and Grzegorz Plebanek

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Ideals on ω

Basic definitions. We consider ideals of subsets of ω. An ideal I is analytic if it is analytic as a subset of 2ω. An ideal I is a P-ideal if for every (An)n from I there is A ∈ I such that An ⊆∗ A for each n. An ideal I is tall if each infinite set contains an infinite element of I.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Ideals on ω

Basic definitions. We consider ideals of subsets of ω. An ideal I is analytic if it is analytic as a subset of 2ω. An ideal I is a P-ideal if for every (An)n from I there is A ∈ I such that An ⊆∗ A for each n. An ideal I is tall if each infinite set contains an infinite element of I.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Ideals on ω

Basic definitions. We consider ideals of subsets of ω. An ideal I is analytic if it is analytic as a subset of 2ω. An ideal I is a P-ideal if for every (An)n from I there is A ∈ I such that An ⊆∗ A for each n. An ideal I is tall if each infinite set contains an infinite element of I.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Ideals on ω

Basic definitions. We consider ideals of subsets of ω. An ideal I is analytic if it is analytic as a subset of 2ω. An ideal I is a P-ideal if for every (An)n from I there is A ∈ I such that An ⊆∗ A for each n. An ideal I is tall if each infinite set contains an infinite element of I.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Ideals on ω

Basic definitions. We consider ideals of subsets of ω. An ideal I is analytic if it is analytic as a subset of 2ω. An ideal I is a P-ideal if for every (An)n from I there is A ∈ I such that An ⊆∗ A for each n. An ideal I is tall if each infinite set contains an infinite element of I.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Summable ideals

Definition: Summable ideals. Consider a sequence (xn)n from [0, ∞). Say that I belongs to I if and only if

• i∈I

xi < ∞. This kind of ideals are called summable ideals. The summable ideal is defined by xi = 1/i. Remark: It will be convenient to assume that

i xi = ∞ (and

thus I is proper) and that limi→∞ xi = 0 (and thus I is tall).

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Summable ideals

Definition: Summable ideals. Consider a sequence (xn)n from [0, ∞). Say that I belongs to I if and only if

• i∈I

xi < ∞. This kind of ideals are called summable ideals. The summable ideal is defined by xi = 1/i. Remark: It will be convenient to assume that

i xi = ∞ (and

thus I is proper) and that limi→∞ xi = 0 (and thus I is tall).

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Summable ideals

Definition: Summable ideals. Consider a sequence (xn)n from [0, ∞). Say that I belongs to I if and only if

• i∈I

xi < ∞. This kind of ideals are called summable ideals. The summable ideal is defined by xi = 1/i. Remark: It will be convenient to assume that

i xi = ∞ (and

thus I is proper) and that limi→∞ xi = 0 (and thus I is tall).

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Summable ideals

Definition: Summable ideals. Consider a sequence (xn)n from [0, ∞). Say that I belongs to I if and only if

• i∈I

xi < ∞. This kind of ideals are called summable ideals. The summable ideal is defined by xi = 1/i. Remark: It will be convenient to assume that

i xi = ∞ (and

thus I is proper) and that limi→∞ xi = 0 (and thus I is tall).

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Density ideals

Definition: Density ideals. The asymptotic density ideal is defined by A ∈ Z ⇐ ⇒ lim sup

n→∞

|A ∩ n| n = 0.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Density ideals

Definition: Density ideals. The asymptotic density ideal is defined by A ∈ Z ⇐ ⇒ lim sup

n→∞

|A ∩ n| n = 0.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Solecki’s theorem

Theorem Assume I is an analytic P-ideal. There is a lower semi-continuous submeasure ϕ: P(ω) → [0, ∞) such that I = Exh(ϕ). Exh(ϕ) = {A ⊆ ω: lim

n→∞ ϕ(A \ n) = 0}.

If I is Fσ, then there is a LSC submeasure ϕ such that I = Fin(ϕ) (= Exh(ϕ)). Fin(ϕ) = {A ⊆ ω: ϕ(A) < ∞}.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Solecki’s theorem

Theorem Assume I is an analytic P-ideal. There is a lower semi-continuous submeasure ϕ: P(ω) → [0, ∞) such that I = Exh(ϕ). Exh(ϕ) = {A ⊆ ω: lim

n→∞ ϕ(A \ n) = 0}.

If I is Fσ, then there is a LSC submeasure ϕ such that I = Fin(ϕ) (= Exh(ϕ)). Fin(ϕ) = {A ⊆ ω: ϕ(A) < ∞}.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Solecki’s theorem

Theorem Assume I is an analytic P-ideal. There is a lower semi-continuous submeasure ϕ: P(ω) → [0, ∞) such that I = Exh(ϕ). Exh(ϕ) = {A ⊆ ω: lim

n→∞ ϕ(A \ n) = 0}.

If I is Fσ, then there is a LSC submeasure ϕ such that I = Fin(ϕ) (= Exh(ϕ)). Fin(ϕ) = {A ⊆ ω: ϕ(A) < ∞}.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Solecki’s theorem

Theorem Assume I is an analytic P-ideal. There is a lower semi-continuous submeasure ϕ: P(ω) → [0, ∞) such that I = Exh(ϕ). Exh(ϕ) = {A ⊆ ω: lim

n→∞ ϕ(A \ n) = 0}.

If I is Fσ, then there is a LSC submeasure ϕ such that I = Fin(ϕ) (= Exh(ϕ)). Fin(ϕ) = {A ⊆ ω: ϕ(A) < ∞}.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Solecki’s theorem

Theorem Assume I is an analytic P-ideal. There is a lower semi-continuous submeasure ϕ: P(ω) → [0, ∞) such that I = Exh(ϕ). Exh(ϕ) = {A ⊆ ω: lim

n→∞ ϕ(A \ n) = 0}.

If I is Fσ, then there is a LSC submeasure ϕ such that I = Fin(ϕ) (= Exh(ϕ)). Fin(ϕ) = {A ⊆ ω: ϕ(A) < ∞}.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Main idea

Main idea Instead of [0, ∞) in the definition of summable ideals consider

• ther structure.

Abelian topological groups, Polish groups, Banach spaces.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Main idea

Main idea Instead of [0, ∞) in the definition of summable ideals consider

• ther structure.

Abelian topological groups, Polish groups, Banach spaces.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Main idea

Main idea Instead of [0, ∞) in the definition of summable ideals consider

• ther structure.

Abelian topological groups, Polish groups, Banach spaces.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Main idea

Main idea Instead of [0, ∞) in the definition of summable ideals consider

• ther structure.

Abelian topological groups, Polish groups, Banach spaces.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Main idea

Main idea Instead of [0, ∞) in the definition of summable ideals consider

• ther structure.

Abelian topological groups, Polish groups, Banach spaces.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation of ideals

The main definition Let G be an (appropriate, complete) structure. Consider a sequence (xn)n in G such that

• n

xn is not unconditionally convergent. Define IG

(xn) by

A ∈ IG

(xn) ⇐

⇒

• n∈A

xn is unconditionally convergent. If an ideal I is of the form IG

(xn) for some (xn)n in G, then we say

that I is represented in G.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation of ideals

The main definition Let G be an (appropriate, complete) structure. Consider a sequence (xn)n in G such that

• n

xn is not unconditionally convergent. Define IG

(xn) by

A ∈ IG

(xn) ⇐

⇒

• n∈A

xn is unconditionally convergent. If an ideal I is of the form IG

(xn) for some (xn)n in G, then we say

that I is represented in G.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in groups

Proposition Every ideal on ω is represented in a group. Theorem An ideal is represented in a Polish group if and only if it is an analytic P-ideal.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in groups

Proposition Every ideal on ω is represented in a group. Theorem An ideal is represented in a Polish group if and only if it is an analytic P-ideal.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in Banach spaces

Theorem Let I be an ideal. The following are equivalent: I can be represented in a Banach space. I can be represented in ℓ∞. I is a non-pathological analytic P-ideal. Definition: Non-pathological ideals. A submeasure ϕ is non-pathological if ϕ(A) = sup{µ(A): µ is a measure and µ ≤ ϕ} for every A ⊆ ω. An ideal I is non-pathological if I = Exh(ϕ) for a non-pathological submeasure ϕ.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in Banach spaces

Theorem Let I be an ideal. The following are equivalent: I can be represented in a Banach space. I can be represented in ℓ∞. I is a non-pathological analytic P-ideal. Definition: Non-pathological ideals. A submeasure ϕ is non-pathological if ϕ(A) = sup{µ(A): µ is a measure and µ ≤ ϕ} for every A ⊆ ω. An ideal I is non-pathological if I = Exh(ϕ) for a non-pathological submeasure ϕ.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in Banach spaces

Theorem Let I be an ideal. The following are equivalent: I can be represented in a Banach space. I can be represented in ℓ∞. I is a non-pathological analytic P-ideal. Definition: Non-pathological ideals. A submeasure ϕ is non-pathological if ϕ(A) = sup{µ(A): µ is a measure and µ ≤ ϕ} for every A ⊆ ω. An ideal I is non-pathological if I = Exh(ϕ) for a non-pathological submeasure ϕ.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in Banach spaces

Theorem Let I be an ideal. The following are equivalent: I can be represented in a Banach space. I can be represented in ℓ∞. I is a non-pathological analytic P-ideal. Definition: Non-pathological ideals. A submeasure ϕ is non-pathological if ϕ(A) = sup{µ(A): µ is a measure and µ ≤ ϕ} for every A ⊆ ω. An ideal I is non-pathological if I = Exh(ϕ) for a non-pathological submeasure ϕ.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in Banach spaces

Theorem Let I be an ideal. The following are equivalent: I can be represented in a Banach space. I can be represented in ℓ∞. I is a non-pathological analytic P-ideal. Definition: Non-pathological ideals. A submeasure ϕ is non-pathological if ϕ(A) = sup{µ(A): µ is a measure and µ ≤ ϕ} for every A ⊆ ω. An ideal I is non-pathological if I = Exh(ϕ) for a non-pathological submeasure ϕ.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in Banach spaces

Theorem Let I be an ideal. The following are equivalent: I can be represented in a Banach space. I can be represented in ℓ∞. I is a non-pathological analytic P-ideal. Definition: Non-pathological ideals. A submeasure ϕ is non-pathological if ϕ(A) = sup{µ(A): µ is a measure and µ ≤ ϕ} for every A ⊆ ω. An ideal I is non-pathological if I = Exh(ϕ) for a non-pathological submeasure ϕ.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in ℓ∞

I non-pathological = ⇒ I represented in ℓ∞. There is a set {µn : n ∈ ω} of measures such that ϕ(F) = sup µn(F) for every finite F. By lower semi-continuity ϕ(A) = sup µn(A) for every A ⊆ ω. Define xn = (µ0({n}), µ1({n}), . . .).

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in ℓ∞

I non-pathological = ⇒ I represented in ℓ∞. There is a set {µn : n ∈ ω} of measures such that ϕ(F) = sup µn(F) for every finite F. By lower semi-continuity ϕ(A) = sup µn(A) for every A ⊆ ω. Define xn = (µ0({n}), µ1({n}), . . .).

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in ℓ∞

I non-pathological = ⇒ I represented in ℓ∞. There is a set {µn : n ∈ ω} of measures such that ϕ(F) = sup µn(F) for every finite F. By lower semi-continuity ϕ(A) = sup µn(A) for every A ⊆ ω. Define xn = (µ0({n}), µ1({n}), . . .).

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in ℓ∞

xn = (µ0({n}), µ1({n}), . . .). Notice that for each finite F ⊆ ω

• i∈F

xi = (µ0(F), µ1(F), · · · ) and thus

• i∈F

xi = ϕ(F).

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in ℓ∞

xn = (µ0({n}), µ1({n}), . . .). Notice that for each finite F ⊆ ω

• i∈F

xi = (µ0(F), µ1(F), · · · ) and thus

• i∈F

xi = ϕ(F).

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in ℓ∞

I represented in ℓ∞ = ⇒ I non-pathological. Assume I is represented through (xn)n from ℓ∞. WLOG we can assume that xn(k) ≥ 0 for each n and k. Define µn(A) =

• i∈A

xi(n) and notice that each µn is a measure. Let ϕ(A) = sup{µn(A): n ∈ ω} for A ⊆ ω. Again,

• i∈F

xi = ϕ(F). for every finite F ⊆ ω.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in ℓ∞

I represented in ℓ∞ = ⇒ I non-pathological. Assume I is represented through (xn)n from ℓ∞. WLOG we can assume that xn(k) ≥ 0 for each n and k. Define µn(A) =

• i∈A

xi(n) and notice that each µn is a measure. Let ϕ(A) = sup{µn(A): n ∈ ω} for A ⊆ ω. Again,

• i∈F

xi = ϕ(F). for every finite F ⊆ ω.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in ℓ∞

I represented in ℓ∞ = ⇒ I non-pathological. Assume I is represented through (xn)n from ℓ∞. WLOG we can assume that xn(k) ≥ 0 for each n and k. Define µn(A) =

• i∈A

xi(n) and notice that each µn is a measure. Let ϕ(A) = sup{µn(A): n ∈ ω} for A ⊆ ω. Again,

• i∈F

xi = ϕ(F). for every finite F ⊆ ω.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in ℓ∞

I represented in ℓ∞ = ⇒ I non-pathological. Assume I is represented through (xn)n from ℓ∞. WLOG we can assume that xn(k) ≥ 0 for each n and k. Define µn(A) =

• i∈A

xi(n) and notice that each µn is a measure. Let ϕ(A) = sup{µn(A): n ∈ ω} for A ⊆ ω. Again,

• i∈F

xi = ϕ(F). for every finite F ⊆ ω.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in ℓ∞

I represented in ℓ∞ = ⇒ I non-pathological. Assume I is represented through (xn)n from ℓ∞. WLOG we can assume that xn(k) ≥ 0 for each n and k. Define µn(A) =

• i∈A

xi(n) and notice that each µn is a measure. Let ϕ(A) = sup{µn(A): n ∈ ω} for A ⊆ ω. Again,

• i∈F

xi = ϕ(F). for every finite F ⊆ ω.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in ℓ∞

I represented in ℓ∞ = ⇒ I non-pathological. Assume I is represented through (xn)n from ℓ∞. WLOG we can assume that xn(k) ≥ 0 for each n and k. Define µn(A) =

• i∈A

xi(n) and notice that each µn is a measure. Let ϕ(A) = sup{µn(A): n ∈ ω} for A ⊆ ω. Again,

• i∈F

xi = ϕ(F). for every finite F ⊆ ω.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Asymptotic density ideal Remark: A ∈ Z ⇐ ⇒ lim sup

n

|A ∩ [2n, 2n+1)| 2n = 0. Representation: xn(k) = 1 2k if n ∈ [2k, 2k+1), xn(k) = 0 otherwise.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Asymptotic density ideal Remark: A ∈ Z ⇐ ⇒ lim sup

n

|A ∩ [2n, 2n+1)| 2n = 0. Representation: xn(k) = 1 2k if n ∈ [2k, 2k+1), xn(k) = 0 otherwise.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Asymptotic density ideal Remark: A ∈ Z ⇐ ⇒ lim sup

n

|A ∩ [2n, 2n+1)| 2n = 0. Representation: xn(k) = 1 2k if n ∈ [2k, 2k+1), xn(k) = 0 otherwise.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Trace of null Definition: tr(N) = {A ⊆ 2<ω : {x ∈ 2ω : ∃∞n x|n ∈ A} ∈ N} Representation: ... in ℓ∞(F), where F is the set of all finite antichains in 2<ω. The sequence will be indexed by 2<ω. xs(F) = 1/2|s| if s ∈ F xs(F) = 0 otherwise.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Trace of null Definition: tr(N) = {A ⊆ 2<ω : {x ∈ 2ω : ∃∞n x|n ∈ A} ∈ N} Representation: ... in ℓ∞(F), where F is the set of all finite antichains in 2<ω. The sequence will be indexed by 2<ω. xs(F) = 1/2|s| if s ∈ F xs(F) = 0 otherwise.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Trace of null Definition: tr(N) = {A ⊆ 2<ω : {x ∈ 2ω : ∃∞n x|n ∈ A} ∈ N} Representation: ... in ℓ∞(F), where F is the set of all finite antichains in 2<ω. The sequence will be indexed by 2<ω. xs(F) = 1/2|s| if s ∈ F xs(F) = 0 otherwise.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Trace of null Definition: tr(N) = {A ⊆ 2<ω : {x ∈ 2ω : ∃∞n x|n ∈ A} ∈ N} Representation: ... in ℓ∞(F), where F is the set of all finite antichains in 2<ω. The sequence will be indexed by 2<ω. xs(F) = 1/2|s| if s ∈ F xs(F) = 0 otherwise.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Summable ideal (alternative representation) Remark: A ∈ S ⇐ ⇒

• n

A ∩ [2n, 2n+1) 2n < ∞. “∞-dimensional” representation: ... in ℓ∞(F), where F is the set of all finite subsets of ω. xn(F) = 1/2k if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Summable ideal (alternative representation) Remark: A ∈ S ⇐ ⇒

• n

A ∩ [2n, 2n+1) 2n < ∞. “∞-dimensional” representation: ... in ℓ∞(F), where F is the set of all finite subsets of ω. xn(F) = 1/2k if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Summable ideal (alternative representation) Remark: A ∈ S ⇐ ⇒

• n

A ∩ [2n, 2n+1) 2n < ∞. “∞-dimensional” representation: ... in ℓ∞(F), where F is the set of all finite subsets of ω. xn(F) = 1/2k if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Summable ideal (alternative representation) Remark: A ∈ S ⇐ ⇒

• n

A ∩ [2n, 2n+1) 2n < ∞. “∞-dimensional” representation: ... in ℓ∞(F), where F is the set of all finite subsets of ω. xn(F) = 1/2k if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Summable ideal (alternative representation) Remark: A ∈ S ⇐ ⇒

• n

A ∩ [2n, 2n+1) 2n < ∞. “∞-dimensional” representation: ... in ℓ∞(F), where F is the set of all finite subsets of ω. xn(F) = 1/2k if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Farah’s ideal (Fσ, non-summable) Definition: A ∈ F ⇐ ⇒

• n

min(n, |[2n, 2n+1) ∩ A|) n2 < ∞. Representation: . . . in ℓ∞(F), where F is the set of all finite subsets F of ω such that |F ∩ [2n, 2n+1)| ≤ n. xn(F) = 1/k2 if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Farah’s ideal (Fσ, non-summable) Definition: A ∈ F ⇐ ⇒

• n

min(n, |[2n, 2n+1) ∩ A|) n2 < ∞. Representation: . . . in ℓ∞(F), where F is the set of all finite subsets F of ω such that |F ∩ [2n, 2n+1)| ≤ n. xn(F) = 1/k2 if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Farah’s ideal (Fσ, non-summable) Definition: A ∈ F ⇐ ⇒

• n

min(n, |[2n, 2n+1) ∩ A|) n2 < ∞. Representation: . . . in ℓ∞(F), where F is the set of all finite subsets F of ω such that |F ∩ [2n, 2n+1)| ≤ n. xn(F) = 1/k2 if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Farah’s ideal (Fσ, non-summable) Definition: A ∈ F ⇐ ⇒

• n

min(n, |[2n, 2n+1) ∩ A|) n2 < ∞. Representation: . . . in ℓ∞(F), where F is the set of all finite subsets F of ω such that |F ∩ [2n, 2n+1)| ≤ n. xn(F) = 1/k2 if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Farah’s ideal (Fσ, non-summable) Definition: A ∈ F ⇐ ⇒

• n

min(n, |[2n, 2n+1) ∩ A|) n2 < ∞. Representation: . . . in ℓ∞(F), where F is the set of all finite subsets F of ω such that |F ∩ [2n, 2n+1)| ≤ n. xn(F) = 1/k2 if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Abstract way Consider a family F of finite subsets of ω and define xn(F) = 1/2k if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F. Define I = {A ⊆ ω:

• n∈A

xn(F) is convergent in ℓ∞(F)}. Which ideals are of this form?

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Abstract way Consider a family F of finite subsets of ω and define xn(F) = 1/2k if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F. Define I = {A ⊆ ω:

• n∈A

xn(F) is convergent in ℓ∞(F)}. Which ideals are of this form?

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Abstract way Consider a family F of finite subsets of ω and define xn(F) = 1/2k if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F. Define I = {A ⊆ ω:

• n∈A

xn(F) is convergent in ℓ∞(F)}. Which ideals are of this form?

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Examples

Abstract way Consider a family F of finite subsets of ω and define xn(F) = 1/2k if n ∈ [2k, 2k+1) ∩ F, xn(F) = 0 if n / ∈ F. Define I = {A ⊆ ω:

• n∈A

xn(F) is convergent in ℓ∞(F)}. Which ideals are of this form?

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Tsirelson ideal

Definition (Farah, Velickovic) Let (ek)k be the standard base of the Tsirelson space. T ∈ T ⇐ ⇒

• n∈A

en · 1/n is convergent in the Tsirelson space. T is Fσ, non-summable.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Tsirelson ideal

Definition (Farah, Velickovic) Let (ek)k be the standard base of the Tsirelson space. T ∈ T ⇐ ⇒

• n∈A

en · 1/n is convergent in the Tsirelson space. T is Fσ, non-summable.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in ℓ1

Fact If I can be represented in ℓ1 through a sequence (xn)n such that xn(k) ≥ 0 for each n and k, then I is a summable ideal. However, there is an ideal represented in ℓ1 which is not summable.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in ℓ1

Fact If I can be represented in ℓ1 through a sequence (xn)n such that xn(k) ≥ 0 for each n and k, then I is a summable ideal. However, there is an ideal represented in ℓ1 which is not summable.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Representation in c0

Question Which ideals can be represented in c0?

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Structure of analytic P-ideals

Density-like ideals An ideal Exh(ϕ) is density-like if ∀ε > 0 ∃δ ∀(Fn)n such that ϕ(Fn) < δ ∃A ⊆ ω ϕ(

• n∈A

Fn) < ε. Summable-like ideals An ideal Exh(ϕ) is summable-like if ∃ε > 0 ∀δ > 0 ∃(Fn)n such that ϕ(Fn) < δ ∃k ∈ ω ∀G ∈ [ω]k ϕ(

• n∈G

Fn) ≥ ε.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Structure of analytic P-ideals

Density-like ideals An ideal Exh(ϕ) is density-like if ∀ε > 0 ∃δ ∀(Fn)n such that ϕ(Fn) < δ ∃A ⊆ ω ϕ(

• n∈A

Fn) < ε. Summable-like ideals An ideal Exh(ϕ) is summable-like if ∃ε > 0 ∀δ > 0 ∃(Fn)n such that ϕ(Fn) < δ ∃k ∈ ω ∀G ∈ [ω]k ϕ(

• n∈G

Fn) ≥ ε.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

tr(N) is not represented in c0.

Assume (xn)n is a c0-representation of tr(N). WLOG {k : xn(k) = 0} is finite for each n. tr(N) is totally bounded. Hence, WLOG {n: xn(k) = 0} is finite for each k. So, tr(N) would be density-like. But tr(N) is summable-like. A contradiction.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

tr(N) is not represented in c0.

Assume (xn)n is a c0-representation of tr(N). WLOG {k : xn(k) = 0} is finite for each n. tr(N) is totally bounded. Hence, WLOG {n: xn(k) = 0} is finite for each k. So, tr(N) would be density-like. But tr(N) is summable-like. A contradiction.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

tr(N) is not represented in c0.

Assume (xn)n is a c0-representation of tr(N). WLOG {k : xn(k) = 0} is finite for each n. tr(N) is totally bounded. Hence, WLOG {n: xn(k) = 0} is finite for each k. So, tr(N) would be density-like. But tr(N) is summable-like. A contradiction.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

tr(N) is not represented in c0.

Assume (xn)n is a c0-representation of tr(N). WLOG {k : xn(k) = 0} is finite for each n. tr(N) is totally bounded. Hence, WLOG {n: xn(k) = 0} is finite for each k. So, tr(N) would be density-like. But tr(N) is summable-like. A contradiction.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

tr(N) is not represented in c0.

Assume (xn)n is a c0-representation of tr(N). WLOG {k : xn(k) = 0} is finite for each n. tr(N) is totally bounded. Hence, WLOG {n: xn(k) = 0} is finite for each k. So, tr(N) would be density-like. But tr(N) is summable-like. A contradiction.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

tr(N) is not represented in c0.

Assume (xn)n is a c0-representation of tr(N). WLOG {k : xn(k) = 0} is finite for each n. tr(N) is totally bounded. Hence, WLOG {n: xn(k) = 0} is finite for each k. So, tr(N) would be density-like. But tr(N) is summable-like. A contradiction.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Fσ non-summable ideals are not represented in c0.

Theorem If I is an Fσ ideal and is not summable, then it is not represented in c0. Corollary Farah’s ideal is not represented in c0. Tsirelson ideals are not represented in c0.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Fσ non-summable ideals are not represented in c0.

Theorem If I is an Fσ ideal and is not summable, then it is not represented in c0. Corollary Farah’s ideal is not represented in c0. Tsirelson ideals are not represented in c0.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Fσ non-summable ideals are not represented in c0.

Theorem If I is an Fσ ideal and is not summable, then it is not represented in c0. Corollary Farah’s ideal is not represented in c0. Tsirelson ideals are not represented in c0.

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Fσ non-summable ideals are not represented in c0.

Assume I = Exh(ϕ) = Fin(ϕ) is not summable and ϕ = sup µn is a c0-representation [cheating!] Assume there is f : ω → ω such that µn(k) = 0 if n > f (k). Let ϕn = maxm≤n µn and In = Exh(ϕn). Let An ∈ In \ I. WLOG (An)n is pairwise disjoint and ϕn(Am) < 1/2n for m ≥ n. (blackboard)

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Fσ non-summable ideals are not represented in c0.

Assume I = Exh(ϕ) = Fin(ϕ) is not summable and ϕ = sup µn is a c0-representation [cheating!] Assume there is f : ω → ω such that µn(k) = 0 if n > f (k). Let ϕn = maxm≤n µn and In = Exh(ϕn). Let An ∈ In \ I. WLOG (An)n is pairwise disjoint and ϕn(Am) < 1/2n for m ≥ n. (blackboard)

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Fσ non-summable ideals are not represented in c0.

Assume I = Exh(ϕ) = Fin(ϕ) is not summable and ϕ = sup µn is a c0-representation [cheating!] Assume there is f : ω → ω such that µn(k) = 0 if n > f (k). Let ϕn = maxm≤n µn and In = Exh(ϕn). Let An ∈ In \ I. WLOG (An)n is pairwise disjoint and ϕn(Am) < 1/2n for m ≥ n. (blackboard)

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Fσ non-summable ideals are not represented in c0.

Assume I = Exh(ϕ) = Fin(ϕ) is not summable and ϕ = sup µn is a c0-representation [cheating!] Assume there is f : ω → ω such that µn(k) = 0 if n > f (k). Let ϕn = maxm≤n µn and In = Exh(ϕn). Let An ∈ In \ I. WLOG (An)n is pairwise disjoint and ϕn(Am) < 1/2n for m ≥ n. (blackboard)

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Fσ non-summable ideals are not represented in c0.

Assume I = Exh(ϕ) = Fin(ϕ) is not summable and ϕ = sup µn is a c0-representation [cheating!] Assume there is f : ω → ω such that µn(k) = 0 if n > f (k). Let ϕn = maxm≤n µn and In = Exh(ϕn). Let An ∈ In \ I. WLOG (An)n is pairwise disjoint and ϕn(Am) < 1/2n for m ≥ n. (blackboard)

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Fσ non-summable ideals are not represented in c0.

Assume I = Exh(ϕ) = Fin(ϕ) is not summable and ϕ = sup µn is a c0-representation [cheating!] Assume there is f : ω → ω such that µn(k) = 0 if n > f (k). Let ϕn = maxm≤n µn and In = Exh(ϕn). Let An ∈ In \ I. WLOG (An)n is pairwise disjoint and ϕn(Am) < 1/2n for m ≥ n. (blackboard)

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Fσ non-summable ideals are not represented in c0.

Assume I = Exh(ϕ) = Fin(ϕ) is not summable and ϕ = sup µn is a c0-representation [cheating!] Assume there is f : ω → ω such that µn(k) = 0 if n > f (k). Let ϕn = maxm≤n µn and In = Exh(ϕn). Let An ∈ In \ I. WLOG (An)n is pairwise disjoint and ϕn(Am) < 1/2n for m ≥ n. (blackboard)

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Questions

Question How to characterize ideals represented in c0? (density-like modulo “trivial” cases?) Question Is there density-like ideal which is not a generalized density ideal? Question Any connections between families of ideals and Banach spaces?

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Questions

Question How to characterize ideals represented in c0? (density-like modulo “trivial” cases?) Question Is there density-like ideal which is not a generalized density ideal? Question Any connections between families of ideals and Banach spaces?

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Questions

Question How to characterize ideals represented in c0? (density-like modulo “trivial” cases?) Question Is there density-like ideal which is not a generalized density ideal? Question Any connections between families of ideals and Banach spaces?

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Questions

Question How to characterize ideals represented in c0? (density-like modulo “trivial” cases?) Question Is there density-like ideal which is not a generalized density ideal? Question Any connections between families of ideals and Banach spaces?

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

Questions

Question How to characterize ideals represented in c0? (density-like modulo “trivial” cases?) Question Is there density-like ideal which is not a generalized density ideal? Question Any connections between families of ideals and Banach spaces?

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals

Setting the stage Generalization of summability Representations in ℓ∞ Representation in other Banach spaces

The picture

Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals