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 2 ω . An ideal I is a P-ideal if for every ( A n ) n from I there is A ∈ I such that A n ⊆ ∗ 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 2 ω . An ideal I is a P-ideal if for every ( A n ) n from I there is A ∈ I such that A n ⊆ ∗ 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 2 ω . An ideal I is a P-ideal if for every ( A n ) n from I there is A ∈ I such that A n ⊆ ∗ 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 2 ω . An ideal I is a P-ideal if for every ( A n ) n from I there is A ∈ I such that A n ⊆ ∗ 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 2 ω . An ideal I is a P-ideal if for every ( A n ) n from I there is A ∈ I such that A n ⊆ ∗ 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 ( x n ) n from [0 , ∞ ). Say that I belongs to I if and only if � x i < ∞ . i ∈ I This kind of ideals are called summable ideals . The summable ideal is defined by x i = 1 / i . Remark: It will be convenient to assume that � i x i = ∞ (and thus I is proper) and that lim i →∞ x i = 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 ( x n ) n from [0 , ∞ ). Say that I belongs to I if and only if � x i < ∞ . i ∈ I This kind of ideals are called summable ideals . The summable ideal is defined by x i = 1 / i . Remark: It will be convenient to assume that � i x i = ∞ (and thus I is proper) and that lim i →∞ x i = 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 ( x n ) n from [0 , ∞ ). Say that I belongs to I if and only if � x i < ∞ . i ∈ I This kind of ideals are called summable ideals . The summable ideal is defined by x i = 1 / i . Remark: It will be convenient to assume that � i x i = ∞ (and thus I is proper) and that lim i →∞ x i = 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 ( x n ) n from [0 , ∞ ). Say that I belongs to I if and only if � x i < ∞ . i ∈ I This kind of ideals are called summable ideals . The summable ideal is defined by x i = 1 / i . Remark: It will be convenient to assume that � i x i = ∞ (and thus I is proper) and that lim i →∞ x i = 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 ∩ n | A ∈ Z ⇐ ⇒ lim sup = 0 . n 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 Density ideals Definition: Density ideals. The asymptotic density ideal is defined by | A ∩ n | A ∈ Z ⇐ ⇒ lim sup = 0 . n 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 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 : n →∞ ϕ ( A \ n ) = 0 } . lim If I is F σ , 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 : n →∞ ϕ ( A \ n ) = 0 } . lim If I is F σ , 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 : n →∞ ϕ ( A \ n ) = 0 } . lim If I is F σ , 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 : n →∞ ϕ ( A \ n ) = 0 } . lim If I is F σ , 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 : n →∞ ϕ ( A \ n ) = 0 } . lim If I is F σ , 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 other 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 other 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 other structure. Abelian topological groups, Polish groups, Banach spaces. Piotr Borodulin-Nadzieja Vienna 2013 Geometry of analytic P-ideals