Cardinal invariants of the continuum and convergence in dual Banach - - PowerPoint PPT Presentation

Cardinal invariants of the continuum and convergence in dual Banach spaces

Damian Sobota

Institute of Mathematics, Polish Academy of Sciences

Transfinite Methods in Banach Spaces and Operator Algebras

Ideology

Take a classical theorem on weak(*) convergence in a B. space

Ideology

Take a classical theorem on weak(*) convergence in a B. space



• Find substructures of ℘(ω) related to the theorem

Ideology

Take a classical theorem on weak(*) convergence in a B. space



• Find substructures of ℘(ω) related to the theorem



• Assign a cardinal invariant to the structures

Ideology

Take a classical theorem on weak(*) convergence in a B. space



• Find substructures of ℘(ω) related to the theorem



• Assign a cardinal invariant to the structures



• Find lower and upper bounds for the invariant

Ideology

Take a classical theorem on weak(*) convergence in a B. space



• Find substructures of ℘(ω) related to the theorem



• Assign a cardinal invariant to the structures



• Find lower and upper bounds for the invariant



• Obtain independence results

Schur’s theorem

Theorem (Schur, 1921) Every weakly convergent sequence in ℓ1 is norm convergent.

Schur’s theorem

Theorem (Schur, 1921) Every weakly convergent sequence in ℓ1 is norm convergent. Remark on the proof To determine the norm convergence of (xn)n ⊆ ℓ1, it is enough to look at the convergence of the sequence: xn, χA =

• j∈A

xn(j) for every A ∈ ℘(ω).

Schur’s theorem

Theorem (Schur, 1921) Every weakly convergent sequence in ℓ1 is norm convergent. Remark on the proof To determine the norm convergence of (xn)n ⊆ ℓ1, it is enough to look at the convergence of the sequence: xn, χA =

• j∈A

xn(j) for every A ∈ ℘(ω). Definition A family F ⊆ ℘(ω) is Schur if for every sequence (xn)n∈ω ⊆ ℓ1 such that xn, χA → 0 for every A ∈ F, we have lim

n

• xn
• 1 = 0.

Schur number

Definition The Schur number schur is the minimal size of a Schur family: schur = min

|F|: F ⊆ ℘(ω) is Schur .

The pseudo-intersection number

Theorem Assume MAκ(σ-centered) for some cardinal number κ. Then, if F ⊆ ℘(ω) is a Schur family, then |F| > κ.

The pseudo-intersection number

Theorem Assume MAκ(σ-centered) for some cardinal number κ. Then, if F ⊆ ℘(ω) is a Schur family, then |F| > κ. Definition A family F ⊆ [ω]ω has the strong finite intersection property (the SFIP) if G is infinite for every finite G ⊆ F. A set A ∈ [ω]ω is a pseudo-intersecton of F if A \ B is finite for every B ∈ F. p = min

|F|: F ⊆ [ω]ω has SFIP but no pseudo-intersection

The pseudo-intersection number

Theorem Assume MAκ(σ-centered) for some cardinal number κ. Then, if F ⊆ ℘(ω) is a Schur family, then |F| > κ. Definition A family F ⊆ [ω]ω has the strong finite intersection property (the SFIP) if G is infinite for every finite G ⊆ F. A set A ∈ [ω]ω is a pseudo-intersecton of F if A \ B is finite for every B ∈ F. p = min

|F|: F ⊆ [ω]ω has SFIP but no pseudo-intersection

• Theorem (Bell, 1981)

p > κ if and only if MAκ(σ-centered) holds.

Bounds for schur

Theorem

1 Every Schur family is of cardinality at least p.

Bounds for schur

Theorem

1 Every Schur family is of cardinality at least p. 2 Under Martin’s axiom, every Schur family is of cardinality c.

Bounds for schur

Theorem

1 Every Schur family is of cardinality at least p. 2 Under Martin’s axiom, every Schur family is of cardinality c.

Definition (Cofinality of measure) N denotes the Lebesgue null ideal cof(N) = min

|F|: F ⊆ N & ∀A ∈ N∃B ∈ F : A ⊆ B

Bounds for schur

Theorem

1 Every Schur family is of cardinality at least p. 2 Under Martin’s axiom, every Schur family is of cardinality c.

Definition (Cofinality of measure) N denotes the Lebesgue null ideal cof(N) = min

|F|: F ⊆ N & ∀A ∈ N∃B ∈ F : A ⊆ B

• Theorem

There exists a Schur family of cardinality cof(N). Corollary p schur cof(N).

Rosenthal’s lemma

Theorem (Rosenthal, 1970) Let (an)n be an antichain in ℘(ω). Assume (µk)k is a sequence of positive finitely additive measures on ℘(ω) satisfying the inequality µk (

n∈ω an) < 1 for every k ∈ ω. Fix ε > 0.

Rosenthal’s lemma

Theorem (Rosenthal, 1970) Let (an)n be an antichain in ℘(ω). Assume (µk)k is a sequence of positive finitely additive measures on ℘(ω) satisfying the inequality µk (

n∈ω an) < 1 for every k ∈ ω. Fix ε > 0.Then, there exists an

infinite set A ⊆ ω such that for every k ∈ A: µk

• n∈A,n=k

an

• < ε.

Rosenthal’s lemma

Theorem (Rosenthal, 1970) Let (an)n be an antichain in ℘(ω). Assume (µk)k is a sequence of positive finitely additive measures on ℘(ω) satisfying the inequality µk (

n∈ω an) < 1 for every k ∈ ω. Fix ε > 0.Then, there exists an

infinite set A ⊆ ω such that for every k ∈ A: µk

• n∈A,n=k

an

• < ε.

Definition Let F ⊆ [ω]ω. F is called Rosenthal if for every antichain (an)n in ℘(ω), sequence (µk)k of positive measures on ω such that µk (

n∈ω an) < 1 for every k ∈ ω, and ε > 0, there is A ∈ F such

that for every k ∈ A: µk

• n∈A,n=k

an

• < ε.

Rosenthal families

Definition (The Rosenthal number) ros = min

|F|: F ⊆ [ω]ω is Rosenthal

• Theorem

Assume MAκ(countable) for some cardinal number κ. Then, if F ⊆ [ω]ω is a Rosenthal family, then |F| > κ.

Rosenthal families

Definition (The Rosenthal number) ros = min

|F|: F ⊆ [ω]ω is Rosenthal

• Theorem

Assume MAκ(countable) for some cardinal number κ. Then, if F ⊆ [ω]ω is a Rosenthal family, then |F| > κ. Definition (Covering of category) M denotes the ideal of meager subsets of R cov(M) = min

|F|: F ⊆ M covers R

Rosenthal families

Definition (The Rosenthal number) ros = min

|F|: F ⊆ [ω]ω is Rosenthal

• Theorem

Assume MAκ(countable) for some cardinal number κ. Then, if F ⊆ [ω]ω is a Rosenthal family, then |F| > κ. Definition (Covering of category) M denotes the ideal of meager subsets of R cov(M) = min

|F|: F ⊆ M covers R

• Theorem (Keremedis, 1995)

cov(M) > κ if and only if MAκ(countable) holds.

Rosenthal families

Definition (The Rosenthal number) ros = min

|F|: F ⊆ [ω]ω is Rosenthal

• Theorem

Assume MAκ(countable) for some cardinal number κ. Then, if F ⊆ [ω]ω is a Rosenthal family, then |F| > κ. Definition (Covering of category) M denotes the ideal of meager subsets of R cov(M) = min

|F|: F ⊆ M covers R

• Theorem (Keremedis, 1995)

cov(M) > κ if and only if MAκ(countable) holds. Theorem Every Rosenthal family is of cardinality at least cov(M).

Selective ultrafilters on ω

Definition Let F ⊆ [ω]ω be a non-principal ultrafilter. F is selective (also Ramsey) if for every partition ω =

k∈ω Nk (Nk ∈ ℘(ω) \ F)

there is F ∈ F such that |F ∩ Nk| = 1 for every k ∈ ω.

Selective ultrafilters on ω

Definition Let F ⊆ [ω]ω be a non-principal ultrafilter. F is selective (also Ramsey) if for every partition ω =

k∈ω Nk (Nk ∈ ℘(ω) \ F)

there is F ∈ F such that |F ∩ Nk| = 1 for every k ∈ ω. Theorem (Rudin, 1956) Assuming CH, there is a selective ultrafilter. Theorem (Kunen, 1972; Shelah, 1982) There is a model of ZFC without selective ultrafilters.

The selective ultrafilter number

Theorem Assume U is a base of a selective ultrafilter. Then, U is Rosenthal.

The selective ultrafilter number

Theorem Assume U is a base of a selective ultrafilter. Then, U is Rosenthal. Definition (The selective ultrafilter number) us = min

|U| : U is a base of a selective ultrafilter

• Theorem (Baumgartner and Laver, 1979)

There is a model of ZFC in which us = ω1 < ω2 = c.

The selective ultrafilter number

Theorem Assume U is a base of a selective ultrafilter. Then, U is Rosenthal. Definition (The selective ultrafilter number) us = min

|U| : U is a base of a selective ultrafilter

• Theorem (Baumgartner and Laver, 1979)

There is a model of ZFC in which us = ω1 < ω2 = c. Theorem cov(M) ros us.

The Nikodym and Grothendieck properties

Definition Let A be a Boolean algebra. Then, A has: the Nikodym property if every sequence (µn)n of measures

• n A such that µn(a) → 0 for every a ∈ A converges in the

weak* topology;

The Nikodym and Grothendieck properties

Definition Let A be a Boolean algebra. Then, A has: the Nikodym property if every sequence (µn)n of measures

• n A such that µn(a) → 0 for every a ∈ A converges in the

weak* topology; the Grothendieck property if every weak* convergent sequence (µn)n of measures on A is weakly convergent.

The Nikodym and Grothendieck properties

Definition Let A be a Boolean algebra. Then, A has: the Nikodym property if every sequence (µn)n of measures

• n A such that µn(a) → 0 for every a ∈ A converges in the

weak* topology; the Grothendieck property if every weak* convergent sequence (µn)n of measures on A is weakly convergent. Theorem (Nikodym, 1933) If A is a σ-complete Boolean algebra, then A has the Nikodym property. Theorem (Grothendieck, 1956) If A is a σ-complete Boolean algebra, then A has the Grothendieck property.

The Nikodym and Grothendieck numbers

If A has the Nikodym or Grothendieck property, then the Stone space St(A) does not have any non-trivial convergent sequences. So no countable A has the Nikodym or Grothendieck property.

The Nikodym and Grothendieck numbers

If A has the Nikodym or Grothendieck property, then the Stone space St(A) does not have any non-trivial convergent sequences. So no countable A has the Nikodym or Grothendieck property. Definition The Nikodym number nik is defined as follows: nik = min

|A|: infinite A ⊆ ℘(ω) has the Nikodym property .

The Grothendieck number gr is defined as follows: gr = min

|A|: infinite A ⊆ ℘(ω) has the Grothendieck property .

Lower bounds for nik and gr

Definition (The splitting number) F ⊆ [ω]ω is splitting if for every A ∈ [ω]ω there exists B ∈ F such that: A ∩ B ∈ [ω]ω and A \ B ∈ [ω]ω. s = min{|F| : F ⊆ [ω]ω is splitting}.

Lower bounds for nik and gr

Definition (The splitting number) F ⊆ [ω]ω is splitting if for every A ∈ [ω]ω there exists B ∈ F such that: A ∩ B ∈ [ω]ω and A \ B ∈ [ω]ω. s = min{|F| : F ⊆ [ω]ω is splitting}. Definition (The bounding number) F ⊆ ωω is unbounded if there is no f ∈ ωω dominating eventually every g ∈ F. b = min{|F| : F ⊆ ωω is unbounded}.

Lower bounds for nik and gr

Definition (The splitting number) F ⊆ [ω]ω is splitting if for every A ∈ [ω]ω there exists B ∈ F such that: A ∩ B ∈ [ω]ω and A \ B ∈ [ω]ω. s = min{|F| : F ⊆ [ω]ω is splitting}. Definition (The bounding number) F ⊆ ωω is unbounded if there is no f ∈ ωω dominating eventually every g ∈ F. b = min{|F| : F ⊆ ωω is unbounded}. Theorem

1 nik max

b, s, cov(M) .

2 gr max

s, cov(M) .

Upper bounds for nik and gr

Theorem If κ is a cardinal number such that cof(N) κ = cof([κ]ω), then there exists a Boolean algebra A with the Nikodym property and

• f cardinality κ. Hence, nik κ.

Upper bounds for nik and gr

Theorem If κ is a cardinal number such that cof(N) κ = cof([κ]ω), then there exists a Boolean algebra A with the Nikodym property and

• f cardinality κ. Hence, nik κ.

Theorem If κ is a cardinal number such that max(dg, ros) κ = cof([κ]ω), then there exists a Boolean algebra A with the Grothendieck property and of cardinality κ. Hence, gr κ. dg is a certain number related to the Dieudonn´ e-Grothendieck characterization of weakly compact subsets of the space C

2ω∗,

probably dg cof(N)...

Upper bounds for nik and gr

Theorem If κ is a cardinal number such that cof(N) κ = cof([κ]ω), then there exists a Boolean algebra A with the Nikodym property and

• f cardinality κ. Hence, nik κ.

Theorem If κ is a cardinal number such that max(dg, ros) κ = cof([κ]ω), then there exists a Boolean algebra A with the Grothendieck property and of cardinality κ. Hence, gr κ. dg is a certain number related to the Dieudonn´ e-Grothendieck characterization of weakly compact subsets of the space C

2ω∗,

probably dg cof(N)... Theorem (Brech, 2006) It is consistent that gr = ω1 < ω2 = c.

Independence results

Theorem The existence of the following substructures of ℘(ω) of cardinality strictly less than c is undecidable within ZFC+¬CH: Schur families, Rosenthal families, Boolean algebras with the Nikodym property, Boolean algebras with the Grothendieck property.

Independence results

Theorem The existence of the following substructures of ℘(ω) of cardinality strictly less than c is undecidable within ZFC+¬CH: Schur families, Rosenthal families, Boolean algebras with the Nikodym property, Boolean algebras with the Grothendieck property. Thank you for the attention!