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 ℓ 1 is norm convergent.

Schur’s theorem Theorem (Schur, 1921) Every weakly convergent sequence in ℓ 1 is norm convergent. Remark on the proof To determine the norm convergence of ( x n ) n ⊆ ℓ 1 , it is enough to look at the convergence of the sequence: � � x n , χ A � = x n ( j ) j ∈ A for every A ∈ ℘ ( ω ) .

Schur’s theorem Theorem (Schur, 1921) Every weakly convergent sequence in ℓ 1 is norm convergent. Remark on the proof To determine the norm convergence of ( x n ) n ⊆ ℓ 1 , it is enough to look at the convergence of the sequence: � � x n , χ A � = x n ( j ) j ∈ A for every A ∈ ℘ ( ω ) . Definition A family F ⊆ ℘ ( ω ) is Schur if for every sequence ( x n ) n ∈ ω ⊆ ℓ 1 such that � x n , χ A � → 0 for every A ∈ F , we have � � lim � x n 1 = 0 . � n

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

The pseudo-intersection number Theorem Assume MA κ ( 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 . � |F| : F ⊆ [ ω ] ω has SFIP but no pseudo-intersection � p = min

The pseudo-intersection number Theorem Assume MA κ ( 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 . � |F| : F ⊆ [ ω ] ω has SFIP but no pseudo-intersection � p = min Theorem (Bell, 1981) p > κ if and only if MA κ ( 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 � |F| : F ⊆ N & � ∀ A ∈ N∃ B ∈ F : A ⊆ B �� cof ( N ) = min

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 � |F| : F ⊆ N & � ∀ A ∈ N∃ B ∈ F : A ⊆ B �� cof ( N ) = min Theorem There exists a Schur family of cardinality cof ( N ) . Corollary p � schur � cof ( N ) .

Rosenthal’s lemma Theorem (Rosenthal, 1970) Let ( a n ) n be an antichain in ℘ ( ω ) . Assume ( µ k ) k is a sequence of positive finitely additive measures on ℘ ( ω ) satisfying the inequality µ k ( � n ∈ ω a n ) < 1 for every k ∈ ω . Fix ε > 0 .

Rosenthal’s lemma Theorem (Rosenthal, 1970) Let ( a n ) n be an antichain in ℘ ( ω ) . Assume ( µ k ) k is a sequence of positive finitely additive measures on ℘ ( ω ) satisfying the inequality µ k ( � n ∈ ω a n ) < 1 for every k ∈ ω . Fix ε > 0 .Then, there exists an infinite set A ⊆ ω such that for every k ∈ A : � � � µ k a n < ε. n ∈ A , n � = k

Rosenthal’s lemma Theorem (Rosenthal, 1970) Let ( a n ) n be an antichain in ℘ ( ω ) . Assume ( µ k ) k is a sequence of positive finitely additive measures on ℘ ( ω ) satisfying the inequality µ k ( � n ∈ ω a n ) < 1 for every k ∈ ω . Fix ε > 0 .Then, there exists an infinite set A ⊆ ω such that for every k ∈ A : � � � µ k a n < ε. n ∈ A , n � = k Definition Let F ⊆ [ ω ] ω . F is called Rosenthal if for every antichain ( a n ) n in ℘ ( ω ) , sequence ( µ k ) k of positive measures on ω such that µ k ( � n ∈ ω a n ) < 1 for every k ∈ ω , and ε > 0, there is A ∈ F such that for every k ∈ A : � � � µ k a n < ε. n ∈ A , n � = k

Rosenthal families Definition (The Rosenthal number) � |F| : F ⊆ [ ω ] ω is Rosenthal ros = min � Theorem Assume MA κ ( countable ) for some cardinal number κ . Then, if F ⊆ [ ω ] ω is a Rosenthal family, then |F| > κ .

Rosenthal families Definition (The Rosenthal number) � |F| : F ⊆ [ ω ] ω is Rosenthal ros = min � Theorem Assume MA κ ( 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) � |F| : F ⊆ [ ω ] ω is Rosenthal ros = min � Theorem Assume MA κ ( 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 MA κ ( countable ) holds.

Rosenthal families Definition (The Rosenthal number) � |F| : F ⊆ [ ω ] ω is Rosenthal ros = min � Theorem Assume MA κ ( 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 MA κ ( 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 N k ( N k ∈ ℘ ( ω ) \ F ) there is F ∈ F such that | F ∩ N k | = 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 N k ( N k ∈ ℘ ( ω ) \ F ) there is F ∈ F such that | F ∩ N k | = 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) � |U| : U is a base of a selective ultrafilter � u s = min Theorem (Baumgartner and Laver, 1979) There is a model of ZFC in which u s = ω 1 < ω 2 = c .

The selective ultrafilter number Theorem Assume U is a base of a selective ultrafilter. Then, U is Rosenthal. Definition (The selective ultrafilter number) � |U| : U is a base of a selective ultrafilter � u s = min Theorem (Baumgartner and Laver, 1979) There is a model of ZFC in which u s = ω 1 < ω 2 = c . Theorem cov ( M ) � ros � u s .

The Nikodym and Grothendieck properties Definition Let A be a Boolean algebra. Then, A has: the Nikodym property if every sequence ( µ n ) n of measures on A such that µ n ( 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 ( µ n ) n of measures on A such that µ n ( a ) → 0 for every a ∈ A converges in the weak* topology; the Grothendieck property if every weak* convergent sequence ( µ n ) n of measures on A is weakly convergent.