is c i.e. ((c n kk )) primary?set_theory/banach2016/programme/slides/dow.pdf · outline 1 review of...
TRANSCRIPT
Is `∞/c0 i.e. ((C (N∗), ‖ · ‖)) primary?
Alan Dow
Department of Mathematics and StatisticsUniversity of North Carolina Charlotte
July 19, 2016
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Outline
1 Review of the Drewnowski-Roberts work on the consistency.
2 connections to the question of whether C (N∗)⋃n a∗n≡0 is
complemented
3 more on Velickovic’s forcing extension of PFA andKoszmider’s suggestion,
4 Sketch of proof that PFA (and the forcing extension) impliesC (N∗)⋃
n a∗n≡0 is not complemented
I’d like to mention Cristobal Rodrıguez-Porras thesis topic:
... given the great impact that PFA and some of itsfragments has on P(N)/Fin, it is quite natural to expectthat it will have a strong influence over `∞/c0 too,providing perhaps a similarly elegant theory.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Outline
1 Review of the Drewnowski-Roberts work on the consistency.
2 connections to the question of whether C (N∗)⋃n a∗n≡0 is
complemented
3 more on Velickovic’s forcing extension of PFA andKoszmider’s suggestion,
4 Sketch of proof that PFA (and the forcing extension) impliesC (N∗)⋃
n a∗n≡0 is not complemented
I’d like to mention Cristobal Rodrıguez-Porras thesis topic:
... given the great impact that PFA and some of itsfragments has on P(N)/Fin, it is quite natural to expectthat it will have a strong influence over `∞/c0 too,providing perhaps a similarly elegant theory.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Outline
1 Review of the Drewnowski-Roberts work on the consistency.
2 connections to the question of whether C (N∗)⋃n a∗n≡0 is
complemented
3 more on Velickovic’s forcing extension of PFA andKoszmider’s suggestion,
4 Sketch of proof that PFA (and the forcing extension) impliesC (N∗)⋃
n a∗n≡0 is not complemented
I’d like to mention Cristobal Rodrıguez-Porras thesis topic:
... given the great impact that PFA and some of itsfragments has on P(N)/Fin, it is quite natural to expectthat it will have a strong influence over `∞/c0 too,providing perhaps a similarly elegant theory.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Outline
1 Review of the Drewnowski-Roberts work on the consistency.
2 connections to the question of whether C (N∗)⋃n a∗n≡0 is
complemented
3 more on Velickovic’s forcing extension of PFA andKoszmider’s suggestion,
4 Sketch of proof that PFA (and the forcing extension) impliesC (N∗)⋃
n a∗n≡0 is not complemented
I’d like to mention Cristobal Rodrıguez-Porras thesis topic:
... given the great impact that PFA and some of itsfragments has on P(N)/Fin, it is quite natural to expectthat it will have a strong influence over `∞/c0 too,providing perhaps a similarly elegant theory.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Outline
1 Review of the Drewnowski-Roberts work on the consistency.
2 connections to the question of whether C (N∗)⋃n a∗n≡0 is
complemented
3 more on Velickovic’s forcing extension of PFA andKoszmider’s suggestion,
4 Sketch of proof that PFA (and the forcing extension) impliesC (N∗)⋃
n a∗n≡0 is not complemented
I’d like to mention Cristobal Rodrıguez-Porras thesis topic:
... given the great impact that PFA and some of itsfragments has on P(N)/Fin, it is quite natural to expectthat it will have a strong influence over `∞/c0 too,providing perhaps a similarly elegant theory.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Drewnowski-Roberts
A closed subspace E of a Banach space X is complemented ifX = E + F where F is also a closed subspace E ∩ F = {0}.
And, iff, there is a (projection) P : X → E such that F = ker(P)Leonard-Whitfield (1983 Rocky Mountain)math reviews:
Using a result of H. P. Rosenthal the authors show thatevery infinite-dimensional complemented subspace of`∞/c0 contains an isometric copy of `∞. Moreover,`∞/c0 is isomorphic to its square. This suggests that`∞/c0 is primary, which, the authors note, is still anunsolved question.
just for amusement: in proving`∞/c0 ∼ `∞/c0 + `∞ they begin: “Since βN has weight ℵ1”
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Drewnowski-Roberts
A closed subspace E of a Banach space X is complemented ifX = E + F where F is also a closed subspace E ∩ F = {0}.And, iff, there is a (projection) P : X → E such that F = ker(P)
Leonard-Whitfield (1983 Rocky Mountain)math reviews:
Using a result of H. P. Rosenthal the authors show thatevery infinite-dimensional complemented subspace of`∞/c0 contains an isometric copy of `∞. Moreover,`∞/c0 is isomorphic to its square. This suggests that`∞/c0 is primary, which, the authors note, is still anunsolved question.
just for amusement: in proving`∞/c0 ∼ `∞/c0 + `∞ they begin: “Since βN has weight ℵ1”
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Drewnowski-Roberts
A closed subspace E of a Banach space X is complemented ifX = E + F where F is also a closed subspace E ∩ F = {0}.And, iff, there is a (projection) P : X → E such that F = ker(P)Leonard-Whitfield (1983 Rocky Mountain)math reviews:
Using a result of H. P. Rosenthal the authors show thatevery infinite-dimensional complemented subspace of`∞/c0 contains an isometric copy of `∞. Moreover,`∞/c0 is isomorphic to its square. This suggests that`∞/c0 is primary, which, the authors note, is still anunsolved question.
just for amusement: in proving`∞/c0 ∼ `∞/c0 + `∞ they begin: “Since βN has weight ℵ1”
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Drewnowski-Roberts
A closed subspace E of a Banach space X is complemented ifX = E + F where F is also a closed subspace E ∩ F = {0}.And, iff, there is a (projection) P : X → E such that F = ker(P)Leonard-Whitfield (1983 Rocky Mountain)math reviews:
Using a result of H. P. Rosenthal the authors show thatevery infinite-dimensional complemented subspace of`∞/c0 contains an isometric copy of `∞. Moreover,`∞/c0 is isomorphic to its square. This suggests that`∞/c0 is primary, which, the authors note, is still anunsolved question.
just for amusement: in proving`∞/c0 ∼ `∞/c0 + `∞ they begin: “Since βN has weight ℵ1”
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
For convenience, following Drewnowski-Roberts, C will denote abounded linear isomorphic copy of C (N∗) and C ∼ E will denotebounded linear isomorphic
Definition
C is primary if whenever C = E + F , either E ∼ C or F ∼ C .
Theorem (Drewnowski-Roberts)
If C = E + F , then for one of E ,F (say E ), E ∼ C + E ′
`∞(X ) is the `∞-sum of countably many copies of Banach spaceX .
If {an : n ∈ ω} are disjoint infinite subsets of N, thenC (
⋃a∗n) ∼ `∞(C ); and every f ∈ C (
⋃a∗n) extends into C .
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
For convenience, following Drewnowski-Roberts, C will denote abounded linear isomorphic copy of C (N∗) and C ∼ E will denotebounded linear isomorphic
Definition
C is primary if whenever C = E + F , either E ∼ C or F ∼ C .
Theorem (Drewnowski-Roberts)
If C = E + F , then for one of E ,F (say E ), E ∼ C + E ′
`∞(X ) is the `∞-sum of countably many copies of Banach spaceX .
If {an : n ∈ ω} are disjoint infinite subsets of N, thenC (
⋃a∗n) ∼ `∞(C ); and every f ∈ C (
⋃a∗n) extends into C .
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
For convenience, following Drewnowski-Roberts, C will denote abounded linear isomorphic copy of C (N∗) and C ∼ E will denotebounded linear isomorphic
Definition
C is primary if whenever C = E + F , either E ∼ C or F ∼ C .
Theorem (Drewnowski-Roberts)
If C = E + F , then for one of E ,F (say E ), E ∼ C + E ′
`∞(X ) is the `∞-sum of countably many copies of Banach spaceX .
If {an : n ∈ ω} are disjoint infinite subsets of N, thenC (
⋃a∗n) ∼ `∞(C ); and every f ∈ C (
⋃a∗n) extends into C .
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
For convenience, following Drewnowski-Roberts, C will denote abounded linear isomorphic copy of C (N∗) and C ∼ E will denotebounded linear isomorphic
Definition
C is primary if whenever C = E + F , either E ∼ C or F ∼ C .
Theorem (Drewnowski-Roberts)
If C = E + F , then for one of E ,F (say E ), E ∼ C + E ′
`∞(X ) is the `∞-sum of countably many copies of Banach spaceX .
If {an : n ∈ ω} are disjoint infinite subsets of N, thenC (
⋃a∗n) ∼ `∞(C ); and every f ∈ C (
⋃a∗n) extends into C .
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem (Drewnowski-Roberts)
If C ∼ `∞(C ) and C = E + F with E ∼ C + E ′ thenE ∼ C + E ′ ∼ `∞(C ) + E ′ ∼ `∞(C + E ′ + F ) + E ′ ∼ `∞(C ) ∼ Chence C is primary.
Definition
For a subset R of N∗, let C (N∗)R≡0 be the closed subspace{f ∈ C (N∗) : f [R] = 0}.
Proposition
If C (N∗)⋃ a∗n≡0 is complemented, then `∞(C ) ∼ C (primary).
Proof.
C ∼ `∞(C ) +F ∼ `∞(C ) + `∞(C ) +F ∼ `∞(C ) +C ∼ `∞(C )
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem (Drewnowski-Roberts)
If C ∼ `∞(C ) and C = E + F with E ∼ C + E ′ thenE ∼ C + E ′ ∼ `∞(C ) + E ′ ∼ `∞(C + E ′ + F ) + E ′ ∼ `∞(C ) ∼ Chence C is primary.
Definition
For a subset R of N∗, let C (N∗)R≡0 be the closed subspace{f ∈ C (N∗) : f [R] = 0}.
Proposition
If C (N∗)⋃ a∗n≡0 is complemented, then `∞(C ) ∼ C (primary).
Proof.
C ∼ `∞(C ) +F ∼ `∞(C ) + `∞(C ) +F ∼ `∞(C ) +C ∼ `∞(C )
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem (Drewnowski-Roberts)
If C ∼ `∞(C ) and C = E + F with E ∼ C + E ′ thenE ∼ C + E ′ ∼ `∞(C ) + E ′ ∼ `∞(C + E ′ + F ) + E ′ ∼ `∞(C ) ∼ Chence C is primary.
Definition
For a subset R of N∗, let C (N∗)R≡0 be the closed subspace{f ∈ C (N∗) : f [R] = 0}.
Proposition
If C (N∗)⋃ a∗n≡0 is complemented, then `∞(C ) ∼ C (primary).
Proof.
C ∼ `∞(C ) +F ∼ `∞(C ) + `∞(C ) +F ∼ `∞(C ) +C ∼ `∞(C )
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem (Drewnowski-Roberts)
If C ∼ `∞(C ) and C = E + F with E ∼ C + E ′ thenE ∼ C + E ′ ∼ `∞(C ) + E ′ ∼ `∞(C + E ′ + F ) + E ′ ∼ `∞(C ) ∼ Chence C is primary.
Definition
For a subset R of N∗, let C (N∗)R≡0 be the closed subspace{f ∈ C (N∗) : f [R] = 0}.
Proposition
If C (N∗)⋃ a∗n≡0 is complemented, then `∞(C ) ∼ C (primary).
Proof.
C ∼ `∞(C ) +F ∼ `∞(C ) + `∞(C ) +F ∼ `∞(C ) +C ∼ `∞(C )
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem
If R ⊂ N∗ is a retract (r : N∗ 7→ R, idR ⊂ r), then C (N∗)R≡0 iscomplemented.
Theorem (Negrepontis)
CH implies that⋃a∗n is a retract.
Anti-Theorem⋃a∗n is almost surely not a retract
not in Cohen model (1984), not under PFA (Farah 2000)
Drewnowski-Roberts knew the Cohen model result and stated intheir paper that primary is probably not ZFC, but possiblyequivalent to C ∼ `∞(C )
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem
If R ⊂ N∗ is a retract (r : N∗ 7→ R, idR ⊂ r), then C (N∗)R≡0 iscomplemented.
Theorem (Negrepontis)
CH implies that⋃a∗n is a retract.
Anti-Theorem⋃a∗n is almost surely not a retract
not in Cohen model (1984), not under PFA (Farah 2000)
Drewnowski-Roberts knew the Cohen model result and stated intheir paper that primary is probably not ZFC, but possiblyequivalent to C ∼ `∞(C )
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem
If R ⊂ N∗ is a retract (r : N∗ 7→ R, idR ⊂ r), then C (N∗)R≡0 iscomplemented.
Theorem (Negrepontis)
CH implies that⋃a∗n is a retract.
Anti-Theorem⋃a∗n is almost surely not a retract
not in Cohen model (1984), not under PFA (Farah 2000)
Drewnowski-Roberts knew the Cohen model result and stated intheir paper that primary is probably not ZFC, but possiblyequivalent to C ∼ `∞(C )
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem
If R ⊂ N∗ is a retract (r : N∗ 7→ R, idR ⊂ r), then C (N∗)R≡0 iscomplemented.
Theorem (Negrepontis)
CH implies that⋃a∗n is a retract.
Anti-Theorem⋃a∗n is almost surely not a retract
not in Cohen model (1984), not under PFA (Farah 2000)
Drewnowski-Roberts knew the Cohen model result and stated intheir paper that primary is probably not ZFC, but possiblyequivalent to C ∼ `∞(C )
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Can we use a retract to get a negative decomposition to primary?
Proposition (not promising)
If R is a retract of N∗ such that β(N× R) (i.e. ⊕nRn) is also(homeomorphic to) a retract of N∗, are there ZFC failures?
then C (N∗)R≡0 is isomorphic to C
e.g. C (N∗)x≡0 or C (N∗)R≡0 for any separable retract R.frequently R itself is homeomorphic to β(N× R)
Proof.
let r : N∗ → ⊕nRn.
Each f ∈ C (N∗) uniquely represented f0 + f1with f0 = f ◦ r and f1[⊕nRn] = {0}. The shift strategy: letσ : ⊕nRn → ⊕nRn where σ(Rn) = Rn+1.Then for T (f ) = (f0 ◦ σ) ◦ r + f1 we get T : C (N∗)R0≡0 → C (N∗)we are shifting the values from Rn+1 down to Rn.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Can we use a retract to get a negative decomposition to primary?
Proposition (not promising)
If R is a retract of N∗ such that β(N× R) (i.e. ⊕nRn) is also(homeomorphic to) a retract of N∗, are there ZFC failures?
then C (N∗)R≡0 is isomorphic to C
e.g. C (N∗)x≡0 or C (N∗)R≡0 for any separable retract R.frequently R itself is homeomorphic to β(N× R)
Proof.
let r : N∗ → ⊕nRn.
Each f ∈ C (N∗) uniquely represented f0 + f1with f0 = f ◦ r and f1[⊕nRn] = {0}. The shift strategy: letσ : ⊕nRn → ⊕nRn where σ(Rn) = Rn+1.Then for T (f ) = (f0 ◦ σ) ◦ r + f1 we get T : C (N∗)R0≡0 → C (N∗)we are shifting the values from Rn+1 down to Rn.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Can we use a retract to get a negative decomposition to primary?
Proposition (not promising)
If R is a retract of N∗ such that β(N× R) (i.e. ⊕nRn) is also(homeomorphic to) a retract of N∗, are there ZFC failures?
then C (N∗)R≡0 is isomorphic to C
e.g. C (N∗)x≡0 or C (N∗)R≡0 for any separable retract R.frequently R itself is homeomorphic to β(N× R)
Proof.
let r : N∗ → ⊕nRn.
Each f ∈ C (N∗) uniquely represented f0 + f1with f0 = f ◦ r and f1[⊕nRn] = {0}. The shift strategy: letσ : ⊕nRn → ⊕nRn where σ(Rn) = Rn+1.Then for T (f ) = (f0 ◦ σ) ◦ r + f1 we get T : C (N∗)R0≡0 → C (N∗)we are shifting the values from Rn+1 down to Rn.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Can we use a retract to get a negative decomposition to primary?
Proposition (not promising)
If R is a retract of N∗ such that β(N× R) (i.e. ⊕nRn) is also(homeomorphic to) a retract of N∗, are there ZFC failures?
then C (N∗)R≡0 is isomorphic to C
e.g. C (N∗)x≡0 or C (N∗)R≡0 for any separable retract R.frequently R itself is homeomorphic to β(N× R)
Proof.
let r : N∗ → ⊕nRn. Each f ∈ C (N∗) uniquely represented f0 + f1with f0 = f ◦ r and f1[⊕nRn] = {0}.
The shift strategy: letσ : ⊕nRn → ⊕nRn where σ(Rn) = Rn+1.Then for T (f ) = (f0 ◦ σ) ◦ r + f1 we get T : C (N∗)R0≡0 → C (N∗)we are shifting the values from Rn+1 down to Rn.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Can we use a retract to get a negative decomposition to primary?
Proposition (not promising)
If R is a retract of N∗ such that β(N× R) (i.e. ⊕nRn) is also(homeomorphic to) a retract of N∗, are there ZFC failures?
then C (N∗)R≡0 is isomorphic to C
e.g. C (N∗)x≡0 or C (N∗)R≡0 for any separable retract R.frequently R itself is homeomorphic to β(N× R)
Proof.
let r : N∗ → ⊕nRn. Each f ∈ C (N∗) uniquely represented f0 + f1with f0 = f ◦ r and f1[⊕nRn] = {0}. The shift strategy: letσ : ⊕nRn → ⊕nRn where σ(Rn) = Rn+1.
Then for T (f ) = (f0 ◦ σ) ◦ r + f1 we get T : C (N∗)R0≡0 → C (N∗)we are shifting the values from Rn+1 down to Rn.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Can we use a retract to get a negative decomposition to primary?
Proposition (not promising)
If R is a retract of N∗ such that β(N× R) (i.e. ⊕nRn) is also(homeomorphic to) a retract of N∗, are there ZFC failures?
then C (N∗)R≡0 is isomorphic to C
e.g. C (N∗)x≡0 or C (N∗)R≡0 for any separable retract R.frequently R itself is homeomorphic to β(N× R)
Proof.
let r : N∗ → ⊕nRn. Each f ∈ C (N∗) uniquely represented f0 + f1with f0 = f ◦ r and f1[⊕nRn] = {0}. The shift strategy: letσ : ⊕nRn → ⊕nRn where σ(Rn) = Rn+1.Then for T (f ) = (f0 ◦ σ) ◦ r + f1 we get T : C (N∗)R0≡0 → C (N∗)we are shifting the values from Rn+1 down to Rn.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem (Brecht-Koszmider 2014)
In the Cohen model, not only is⋃a∗n not a retract,
and, not only is `∞(C ) not a complemented subspace of C ,`∞(C ) is not even embeddedable.
Question
Is C (N∗) primary in the Cohen model?If yes, then that answers one Drewnowski-Roberts question.
Questions in the Cohen model
Are there any complemented subspaces of C (N∗) worth examining?
Because, most topological subspaces are themselves copies of N∗.
e.g. Is there a retract R with R 6≈ N∗ and ⊕nRn not a retract?
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem (Brecht-Koszmider 2014)
In the Cohen model, not only is⋃a∗n not a retract,
and, not only is `∞(C ) not a complemented subspace of C ,`∞(C ) is not even embeddedable.
Question
Is C (N∗) primary in the Cohen model?If yes, then that answers one Drewnowski-Roberts question.
Questions in the Cohen model
Are there any complemented subspaces of C (N∗) worth examining?
Because, most topological subspaces are themselves copies of N∗.
e.g. Is there a retract R with R 6≈ N∗ and ⊕nRn not a retract?
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem (Brecht-Koszmider 2014)
In the Cohen model, not only is⋃a∗n not a retract,
and, not only is `∞(C ) not a complemented subspace of C ,`∞(C ) is not even embeddedable.
Question
Is C (N∗) primary in the Cohen model?If yes, then that answers one Drewnowski-Roberts question.
Questions in the Cohen model
Are there any complemented subspaces of C (N∗) worth examining?
Because, most topological subspaces are themselves copies of N∗.
e.g. Is there a retract R with R 6≈ N∗ and ⊕nRn not a retract?
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
so where can we find candidate partitions: C = E + F to testprimary?
Theorem (Velickovic)
There is a poset P2 that PFA implies is ℵ2-distributive and, in theextension, we have N∗ = A⊕U B where (tie-point) like primaryA,B is a closed cover and U is the unique common limit point.
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Corollary
If N∗ = A⊕U B andIA = {a ⊂ N : a∗ ⊂ A} and IB = {b ⊂ N : b∗ ⊂ B}, then
IA ∪ IB generates a dense ideal, one is a P-ideal,OCA fails, and PFA ` P2 forces each are Pℵ2-ideals.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
so where can we find candidate partitions: C = E + F to testprimary?
Theorem (Velickovic)
There is a poset P2 that PFA implies is ℵ2-distributive and, in theextension, we have N∗ = A⊕U B where (tie-point) like primaryA,B is a closed cover and U is the unique common limit point.
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Corollary
If N∗ = A⊕U B andIA = {a ⊂ N : a∗ ⊂ A} and IB = {b ⊂ N : b∗ ⊂ B}, then
IA ∪ IB generates a dense ideal, one is a P-ideal,OCA fails, and PFA ` P2 forces each are Pℵ2-ideals.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
so where can we find candidate partitions: C = E + F to testprimary?
Theorem (Velickovic)
There is a poset P2 that PFA implies is ℵ2-distributive and, in theextension, we have N∗ = A⊕U B where (tie-point) like primaryA,B is a closed cover and U is the unique common limit point.
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UA B
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Corollary
If N∗ = A⊕U B andIA = {a ⊂ N : a∗ ⊂ A} and IB = {b ⊂ N : b∗ ⊂ B}, then
IA ∪ IB generates a dense ideal, one is a P-ideal,OCA fails, and PFA ` P2 forces each are Pℵ2-ideals.
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Koszmider Question
If N∗ = A⊕U B, then (using retracts)
C (N∗) ∼ C (N∗)U≡0 = C (N∗)A≡0 + C (N∗)B≡0
and C (N∗)A≡0 ∼ C (B) and C (N∗)B≡0 ∼ C (A),So, can we get C (A) 6∼ C (N∗) and C (B) 6∼ C (N∗)?
well the first job is to get A,B 6≈ N∗
Theorem (Velickovic)
P2 A ≈ B but we do not know about A ≈ N∗
I was already working on this because, with my PhD student, itcame up in our efforts connected to a van Douwen questionmust a precise 2-to-1 image of N∗ be a copy of N∗?
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Koszmider Question
If N∗ = A⊕U B, then (using retracts)
C (N∗) ∼ C (N∗)U≡0 = C (N∗)A≡0 + C (N∗)B≡0
and C (N∗)A≡0 ∼ C (B) and C (N∗)B≡0 ∼ C (A),So, can we get C (A) 6∼ C (N∗) and C (B) 6∼ C (N∗)?
well the first job is to get A,B 6≈ N∗
Theorem (Velickovic)
P2 A ≈ B but we do not know about A ≈ N∗
I was already working on this because, with my PhD student, itcame up in our efforts connected to a van Douwen questionmust a precise 2-to-1 image of N∗ be a copy of N∗?
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Koszmider Question
If N∗ = A⊕U B, then (using retracts)
C (N∗) ∼ C (N∗)U≡0 = C (N∗)A≡0 + C (N∗)B≡0
and C (N∗)A≡0 ∼ C (B) and C (N∗)B≡0 ∼ C (A),So, can we get C (A) 6∼ C (N∗) and C (B) 6∼ C (N∗)?
well the first job is to get A,B 6≈ N∗
Theorem (Velickovic)
P2 A ≈ B but we do not know about A ≈ N∗
I was already working on this because, with my PhD student, itcame up in our efforts connected to a van Douwen questionmust a precise 2-to-1 image of N∗ be a copy of N∗?
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Koszmider Question
If N∗ = A⊕U B, then (using retracts)
C (N∗) ∼ C (N∗)U≡0 = C (N∗)A≡0 + C (N∗)B≡0
and C (N∗)A≡0 ∼ C (B) and C (N∗)B≡0 ∼ C (A),So, can we get C (A) 6∼ C (N∗) and C (B) 6∼ C (N∗)?
well the first job is to get A,B 6≈ N∗
Theorem (Velickovic)
P2 A ≈ B but we do not know about A ≈ N∗
I was already working on this because, with my PhD student, itcame up in our efforts connected to a van Douwen questionmust a precise 2-to-1 image of N∗ be a copy of N∗?
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Picking up on work of Steprans with the forcing (P2)κ to get fewautomorphisms
Theorem (2009 with Shelah)
Forcing with P2 × P2 over PFA we get
N∗ = (A1 ⊕U1 B1)⊕ (A2 ⊕U2 B2) while
A1 ⊕U1≡U2 B2 6≈ N∗ yet it is a precise 2-to-1 image of N∗.
but?? I still didn’t know if either A1,A2 were copies of N∗
I was inspired by Steprans remark about P2:adding a non-trivial automorphism and nothing else
The methods use PFA non-trivial automorphism arguments toprove properties of the P2 forcing extension
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Picking up on work of Steprans with the forcing (P2)κ to get fewautomorphisms
Theorem (2009 with Shelah)
Forcing with P2 × P2 over PFA we get
N∗ = (A1 ⊕U1 B1)⊕ (A2 ⊕U2 B2) while
A1 ⊕U1≡U2 B2 6≈ N∗ yet it is a precise 2-to-1 image of N∗.
but?? I still didn’t know if either A1,A2 were copies of N∗
I was inspired by Steprans remark about P2:adding a non-trivial automorphism and nothing else
The methods use PFA non-trivial automorphism arguments toprove properties of the P2 forcing extension
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Picking up on work of Steprans with the forcing (P2)κ to get fewautomorphisms
Theorem (2009 with Shelah)
Forcing with P2 × P2 over PFA we get
N∗ = (A1 ⊕U1 B1)⊕ (A2 ⊕U2 B2) while
A1 ⊕U1≡U2 B2 6≈ N∗ yet it is a precise 2-to-1 image of N∗.
but?? I still didn’t know if either A1,A2 were copies of N∗
I was inspired by Steprans remark about P2:adding a non-trivial automorphism and nothing else
The methods use PFA non-trivial automorphism arguments toprove properties of the P2 forcing extension
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
I modified Boban’s P2 to a combinatorially simpler P1
Theorem
PFA implies that P1 forces N∗ = A⊕U B with A 6≈ N∗ 6≈ Band
⋃n a∗n is not a retract
so maybe we are back in the Primary businesscuriously: and nothing else [Steprans] it appeared at firstthat P1 would force no non-trivial automorphisms, but
Theorem
PFA implies that P1 forces for that special new Uif N∗ = A1 ⊕U B1, then A1 6≈ N∗ 6≈ B1, but
there is a W with N∗ = A2 ⊕W B2 ⊕W C2 and B2 ≈W C2 ≈ N∗and so there is a non-trivial automorphism.
there is hope we can get not Primary, but we must start with PFA
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
I modified Boban’s P2 to a combinatorially simpler P1
Theorem
PFA implies that P1 forces N∗ = A⊕U B with A 6≈ N∗ 6≈ Band
⋃n a∗n is not a retract
so maybe we are back in the Primary businesscuriously: and nothing else [Steprans] it appeared at firstthat P1 would force no non-trivial automorphisms, but
Theorem
PFA implies that P1 forces for that special new Uif N∗ = A1 ⊕U B1, then A1 6≈ N∗ 6≈ B1, but
there is a W with N∗ = A2 ⊕W B2 ⊕W C2 and B2 ≈W C2 ≈ N∗and so there is a non-trivial automorphism.
there is hope we can get not Primary, but we must start with PFA
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
I modified Boban’s P2 to a combinatorially simpler P1
Theorem
PFA implies that P1 forces N∗ = A⊕U B with A 6≈ N∗ 6≈ Band
⋃n a∗n is not a retract
so maybe we are back in the Primary businesscuriously: and nothing else [Steprans] it appeared at firstthat P1 would force no non-trivial automorphisms, but
Theorem
PFA implies that P1 forces for that special new Uif N∗ = A1 ⊕U B1, then A1 6≈ N∗ 6≈ B1, but
there is a W with N∗ = A2 ⊕W B2 ⊕W C2 and B2 ≈W C2 ≈ N∗and so there is a non-trivial automorphism.
there is hope we can get not Primary, but we must start with PFA
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
I modified Boban’s P2 to a combinatorially simpler P1
Theorem
PFA implies that P1 forces N∗ = A⊕U B with A 6≈ N∗ 6≈ Band
⋃n a∗n is not a retract
so maybe we are back in the Primary businesscuriously: and nothing else [Steprans] it appeared at firstthat P1 would force no non-trivial automorphisms, but
Theorem
PFA implies that P1 forces for that special new Uif N∗ = A1 ⊕U B1, then A1 6≈ N∗ 6≈ B1, but
there is a W with N∗ = A2 ⊕W B2 ⊕W C2 and B2 ≈W C2 ≈ N∗and so there is a non-trivial automorphism.
there is hope we can get not Primary, but we must start with PFA
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem
PFA implies (and I think forcing with P1) that
C (N∗)⋃ a∗n≡0 is not complemented
the most likely scenario for the possible bad news:
N∗ = A⊕U B and C (N∗)B≡0 ∼ C (N∗)U≡0is to have pairwise disjoint homeomorphic copies An ⊕Un Bn
and to again use the σ shift strategy
AAAAAA�
�����
U0
B0
A0
��������
���
f = 0σ←AAAAAA�
�����
U1
B1
A1
σ←AAAAAA�
�����
U2
B2
A2
• • •
T (f ) · 1Bn ‘=’f · 1Bn+1
T (f ) · 1An ‘=’f · 1An
I hope the proof extends to exclude this
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem
PFA implies (and I think forcing with P1) that
C (N∗)⋃ a∗n≡0 is not complemented
the most likely scenario for the possible bad news:
N∗ = A⊕U B and C (N∗)B≡0 ∼ C (N∗)U≡0is to have pairwise disjoint homeomorphic copies An ⊕Un Bn
and to again use the σ shift strategy
AAAAAA�
�����
U0
B0
A0
��������
���
f = 0σ←AAAAAA�
�����
U1
B1
A1
σ←AAAAAA�
�����
U2
B2
A2
• • •
T (f ) · 1Bn ‘=’f · 1Bn+1
T (f ) · 1An ‘=’f · 1An
I hope the proof extends to exclude this
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem
PFA implies (and I think forcing with P1) that
C (N∗)⋃ a∗n≡0 is not complemented
the most likely scenario for the possible bad news:
N∗ = A⊕U B and C (N∗)B≡0 ∼ C (N∗)U≡0is to have pairwise disjoint homeomorphic copies An ⊕Un Bn
and to again use the σ shift strategy
AAAAAA�
�����
U0
B0
A0
��������
���
f = 0σ←AAAAAA�
�����
U1
B1
A1
σ←AAAAAA�
�����
U2
B2
A2
• • •
T (f ) · 1Bn ‘=’f · 1Bn+1
T (f ) · 1An ‘=’f · 1An
I hope the proof extends to exclude this
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Theorem
PFA implies (and I think forcing with P1) that
C (N∗)⋃ a∗n≡0 is not complemented
the most likely scenario for the possible bad news:
N∗ = A⊕U B and C (N∗)B≡0 ∼ C (N∗)U≡0is to have pairwise disjoint homeomorphic copies An ⊕Un Bn
and to again use the σ shift strategy
AAAAAA�
�����
U0
B0
A0
��������
���
f = 0σ←AAAAAA�
�����
U1
B1
A1
σ←AAAAAA�
�����
U2
B2
A2
• • •
T (f ) · 1Bn ‘=’f · 1Bn+1
T (f ) · 1An ‘=’f · 1An
I hope the proof extends to exclude this
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Some Questions
1 if C (N∗) is primary, is `∞(C ) ∼ C?
2 if `∞(C ) ∼ C , is C (N∗)⋃ a∗n≡0 complemented?
3 is there a complemented pair C (N∗) = E + F that isn’ttopological?
4 Is there any model and any Parovicenko space K such thatC (K ) 6∼ C (N∗)?
5 for example, how about K = ω∗1/U(ω1) (simple P-point)under PFA? or Cohen model?or (N× 2c)∗ ?
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Some Questions
1 if C (N∗) is primary, is `∞(C ) ∼ C?
2 if `∞(C ) ∼ C , is C (N∗)⋃ a∗n≡0 complemented?
3 is there a complemented pair C (N∗) = E + F that isn’ttopological?
4 Is there any model and any Parovicenko space K such thatC (K ) 6∼ C (N∗)?
5 for example, how about K = ω∗1/U(ω1) (simple P-point)under PFA? or Cohen model?or (N× 2c)∗ ?
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Some Questions
1 if C (N∗) is primary, is `∞(C ) ∼ C?
2 if `∞(C ) ∼ C , is C (N∗)⋃ a∗n≡0 complemented?
3 is there a complemented pair C (N∗) = E + F that isn’ttopological?
4 Is there any model and any Parovicenko space K such thatC (K ) 6∼ C (N∗)?
5 for example, how about K = ω∗1/U(ω1) (simple P-point)under PFA? or Cohen model?or (N× 2c)∗ ?
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Some Questions
1 if C (N∗) is primary, is `∞(C ) ∼ C?
2 if `∞(C ) ∼ C , is C (N∗)⋃ a∗n≡0 complemented?
3 is there a complemented pair C (N∗) = E + F that isn’ttopological?
4 Is there any model and any Parovicenko space K such thatC (K ) 6∼ C (N∗)?
5 for example, how about K = ω∗1/U(ω1) (simple P-point)under PFA? or Cohen model?or (N× 2c)∗ ?
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
Some Questions
1 if C (N∗) is primary, is `∞(C ) ∼ C?
2 if `∞(C ) ∼ C , is C (N∗)⋃ a∗n≡0 complemented?
3 is there a complemented pair C (N∗) = E + F that isn’ttopological?
4 Is there any model and any Parovicenko space K such thatC (K ) 6∼ C (N∗)?
5 for example, how about K = ω∗1/U(ω1) (simple P-point)under PFA? or Cohen model?or (N× 2c)∗ ?
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
sketch of no linear lifting from C (⋃a∗n) to C (N∗)
Definition
1 for a subset a ⊂ N, 1a is the characteristic function of a
2 for f , g ∈ C ∗(N), we let f ∼ g denote the relationlim supn |f (n)− g(n)| = 0, (i.e. ‖f − g‖∗∞ = 0)
3 we let T : C ∗(N) 7→ C ∗(N) be a lifting of a linear lifting /projection as follows: for all f , g ∈ C ∗(N),
4 for all real r , T (r · f ) = r · T (f ),
5 T (f + g) ∼ T (f ) + T (g),
6 for all n, T (f ) · 1an ∼ f · 1an ,
7 if, for all n, f · 1an ∼ 0, then T (f ) ∼ 0
8 there is a finite bound to {‖T (f )‖∗∞ : f ∈ C (N, 2)}9 for a ⊂ N, let Ja = {n : |a ∩ an| = ℵ0} (we frequently assume
a ∩ an = ∅ for n /∈ Ja); we say that a ∈ J providing Ja isinfinite
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
sketch of no linear lifting from C (⋃a∗n) to C (N∗)
Definition
1 for a subset a ⊂ N, 1a is the characteristic function of a
2 for f , g ∈ C ∗(N), we let f ∼ g denote the relationlim supn |f (n)− g(n)| = 0, (i.e. ‖f − g‖∗∞ = 0)
3 we let T : C ∗(N) 7→ C ∗(N) be a lifting of a linear lifting /projection as follows: for all f , g ∈ C ∗(N),
4 for all real r , T (r · f ) = r · T (f ),
5 T (f + g) ∼ T (f ) + T (g),
6 for all n, T (f ) · 1an ∼ f · 1an ,
7 if, for all n, f · 1an ∼ 0, then T (f ) ∼ 0
8 there is a finite bound to {‖T (f )‖∗∞ : f ∈ C (N, 2)}9 for a ⊂ N, let Ja = {n : |a ∩ an| = ℵ0} (we frequently assume
a ∩ an = ∅ for n /∈ Ja); we say that a ∈ J providing Ja isinfinite
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
sketch of no linear lifting from C (⋃a∗n) to C (N∗)
Definition
1 for a subset a ⊂ N, 1a is the characteristic function of a
2 for f , g ∈ C ∗(N), we let f ∼ g denote the relationlim supn |f (n)− g(n)| = 0, (i.e. ‖f − g‖∗∞ = 0)
3 we let T : C ∗(N) 7→ C ∗(N) be a lifting of a linear lifting /projection as follows: for all f , g ∈ C ∗(N),
4 for all real r , T (r · f ) = r · T (f ),
5 T (f + g) ∼ T (f ) + T (g),
6 for all n, T (f ) · 1an ∼ f · 1an ,
7 if, for all n, f · 1an ∼ 0, then T (f ) ∼ 0
8 there is a finite bound to {‖T (f )‖∗∞ : f ∈ C (N, 2)}9 for a ⊂ N, let Ja = {n : |a ∩ an| = ℵ0} (we frequently assume
a ∩ an = ∅ for n /∈ Ja); we say that a ∈ J providing Ja isinfinite
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
sketch of no linear lifting from C (⋃a∗n) to C (N∗)
Definition
1 for a subset a ⊂ N, 1a is the characteristic function of a
2 for f , g ∈ C ∗(N), we let f ∼ g denote the relationlim supn |f (n)− g(n)| = 0, (i.e. ‖f − g‖∗∞ = 0)
3 we let T : C ∗(N) 7→ C ∗(N) be a lifting of a linear lifting /projection as follows: for all f , g ∈ C ∗(N),
4 for all real r , T (r · f ) = r · T (f ),
5 T (f + g) ∼ T (f ) + T (g),
6 for all n, T (f ) · 1an ∼ f · 1an ,
7 if, for all n, f · 1an ∼ 0, then T (f ) ∼ 0
8 there is a finite bound to {‖T (f )‖∗∞ : f ∈ C (N, 2)}9 for a ⊂ N, let Ja = {n : |a ∩ an| = ℵ0} (we frequently assume
a ∩ an = ∅ for n /∈ Ja); we say that a ∈ J providing Ja isinfinite
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
sketch of no linear lifting from C (⋃a∗n) to C (N∗)
Definition
1 for a subset a ⊂ N, 1a is the characteristic function of a
2 for f , g ∈ C ∗(N), we let f ∼ g denote the relationlim supn |f (n)− g(n)| = 0, (i.e. ‖f − g‖∗∞ = 0)
3 we let T : C ∗(N) 7→ C ∗(N) be a lifting of a linear lifting /projection as follows: for all f , g ∈ C ∗(N),
4 for all real r , T (r · f ) = r · T (f ),
5 T (f + g) ∼ T (f ) + T (g),
6 for all n, T (f ) · 1an ∼ f · 1an ,
7 if, for all n, f · 1an ∼ 0, then T (f ) ∼ 0
8 there is a finite bound to {‖T (f )‖∗∞ : f ∈ C (N, 2)}9 for a ⊂ N, let Ja = {n : |a ∩ an| = ℵ0} (we frequently assume
a ∩ an = ∅ for n /∈ Ja); we say that a ∈ J providing Ja isinfinite
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
sketch of no linear lifting from C (⋃a∗n) to C (N∗)
Definition
1 for a subset a ⊂ N, 1a is the characteristic function of a
2 for f , g ∈ C ∗(N), we let f ∼ g denote the relationlim supn |f (n)− g(n)| = 0, (i.e. ‖f − g‖∗∞ = 0)
3 we let T : C ∗(N) 7→ C ∗(N) be a lifting of a linear lifting /projection as follows: for all f , g ∈ C ∗(N),
4 for all real r , T (r · f ) = r · T (f ),
5 T (f + g) ∼ T (f ) + T (g),
6 for all n, T (f ) · 1an ∼ f · 1an ,
7 if, for all n, f · 1an ∼ 0, then T (f ) ∼ 0
8 there is a finite bound to {‖T (f )‖∗∞ : f ∈ C (N, 2)}9 for a ⊂ N, let Ja = {n : |a ∩ an| = ℵ0} (we frequently assume
a ∩ an = ∅ for n /∈ Ja); we say that a ∈ J providing Ja isinfinite
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
sketch of no linear lifting from C (⋃a∗n) to C (N∗)
Definition
1 for a subset a ⊂ N, 1a is the characteristic function of a
2 for f , g ∈ C ∗(N), we let f ∼ g denote the relationlim supn |f (n)− g(n)| = 0, (i.e. ‖f − g‖∗∞ = 0)
3 we let T : C ∗(N) 7→ C ∗(N) be a lifting of a linear lifting /projection as follows: for all f , g ∈ C ∗(N),
4 for all real r , T (r · f ) = r · T (f ),
5 T (f + g) ∼ T (f ) + T (g),
6 for all n, T (f ) · 1an ∼ f · 1an ,
7 if, for all n, f · 1an ∼ 0, then T (f ) ∼ 0
8 there is a finite bound to {‖T (f )‖∗∞ : f ∈ C (N, 2)}9 for a ⊂ N, let Ja = {n : |a ∩ an| = ℵ0} (we frequently assume
a ∩ an = ∅ for n /∈ Ja); we say that a ∈ J providing Ja isinfinite
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
sketch of no linear lifting from C (⋃a∗n) to C (N∗)
Definition
1 for a subset a ⊂ N, 1a is the characteristic function of a
2 for f , g ∈ C ∗(N), we let f ∼ g denote the relationlim supn |f (n)− g(n)| = 0, (i.e. ‖f − g‖∗∞ = 0)
3 we let T : C ∗(N) 7→ C ∗(N) be a lifting of a linear lifting /projection as follows: for all f , g ∈ C ∗(N),
4 for all real r , T (r · f ) = r · T (f ),
5 T (f + g) ∼ T (f ) + T (g),
6 for all n, T (f ) · 1an ∼ f · 1an ,
7 if, for all n, f · 1an ∼ 0, then T (f ) ∼ 0
8 there is a finite bound to {‖T (f )‖∗∞ : f ∈ C (N, 2)}9 for a ⊂ N, let Ja = {n : |a ∩ an| = ℵ0} (we frequently assume
a ∩ an = ∅ for n /∈ Ja); we say that a ∈ J providing Ja isinfinite
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
sketch of no linear lifting from C (⋃a∗n) to C (N∗)
Definition
1 for a subset a ⊂ N, 1a is the characteristic function of a
2 for f , g ∈ C ∗(N), we let f ∼ g denote the relationlim supn |f (n)− g(n)| = 0, (i.e. ‖f − g‖∗∞ = 0)
3 we let T : C ∗(N) 7→ C ∗(N) be a lifting of a linear lifting /projection as follows: for all f , g ∈ C ∗(N),
4 for all real r , T (r · f ) = r · T (f ),
5 T (f + g) ∼ T (f ) + T (g),
6 for all n, T (f ) · 1an ∼ f · 1an ,
7 if, for all n, f · 1an ∼ 0, then T (f ) ∼ 0
8 there is a finite bound to {‖T (f )‖∗∞ : f ∈ C (N, 2)}
9 for a ⊂ N, let Ja = {n : |a ∩ an| = ℵ0} (we frequently assumea ∩ an = ∅ for n /∈ Ja); we say that a ∈ J providing Ja isinfinite
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
sketch of no linear lifting from C (⋃a∗n) to C (N∗)
Definition
1 for a subset a ⊂ N, 1a is the characteristic function of a
2 for f , g ∈ C ∗(N), we let f ∼ g denote the relationlim supn |f (n)− g(n)| = 0, (i.e. ‖f − g‖∗∞ = 0)
3 we let T : C ∗(N) 7→ C ∗(N) be a lifting of a linear lifting /projection as follows: for all f , g ∈ C ∗(N),
4 for all real r , T (r · f ) = r · T (f ),
5 T (f + g) ∼ T (f ) + T (g),
6 for all n, T (f ) · 1an ∼ f · 1an ,
7 if, for all n, f · 1an ∼ 0, then T (f ) ∼ 0
8 there is a finite bound to {‖T (f )‖∗∞ : f ∈ C (N, 2)}9 for a ⊂ N, let Ja = {n : |a ∩ an| = ℵ0} (we frequently assume
a ∩ an = ∅ for n /∈ Ja); we say that a ∈ J providing Ja isinfinite
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
the rough plan
is to recursively construct a mod fin increasing sequence{dα : α ∈ ω1} ⊂ J together with functions{hα : α ∈ ω1 + 1} ⊂ C (N, 2) so that for α < β ≤ ω1
1 hβ · 1dα =∗ hα (hence hα · 1dα =∗ hα)
2 there is aα ⊂ dα+1 \ dα in J ,
3 the family{〈aα ∩ T (hα+1)−1(< 1
3) , aα ∩ T (hα+1)−1(> 23)〉 : α ∈ ω1}
is shaping up to have the Hausdorff-Luzin property
4 so that T (hω1) can not(???) exist by (3)
Condition (4) poses a big problem because it basically requires that
T (hω1) · 1aα ∼ T (hα+1) · 1aα
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
the rough plan
is to recursively construct a mod fin increasing sequence{dα : α ∈ ω1} ⊂ J together with functions{hα : α ∈ ω1 + 1} ⊂ C (N, 2) so that for α < β ≤ ω1
1 hβ · 1dα =∗ hα (hence hα · 1dα =∗ hα)
2 there is aα ⊂ dα+1 \ dα in J ,
3 the family{〈aα ∩ T (hα+1)−1(< 1
3) , aα ∩ T (hα+1)−1(> 23)〉 : α ∈ ω1}
is shaping up to have the Hausdorff-Luzin property
4 so that T (hω1) can not(???) exist by (3)
Condition (4) poses a big problem because it basically requires that
T (hω1) · 1aα ∼ T (hα+1) · 1aα
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
the rough plan
is to recursively construct a mod fin increasing sequence{dα : α ∈ ω1} ⊂ J together with functions{hα : α ∈ ω1 + 1} ⊂ C (N, 2) so that for α < β ≤ ω1
1 hβ · 1dα =∗ hα (hence hα · 1dα =∗ hα)
2 there is aα ⊂ dα+1 \ dα in J ,
3 the family{〈aα ∩ T (hα+1)−1(< 1
3) , aα ∩ T (hα+1)−1(> 23)〉 : α ∈ ω1}
is shaping up to have the Hausdorff-Luzin property
4 so that T (hω1) can not(???) exist by (3)
Condition (4) poses a big problem because it basically requires that
T (hω1) · 1aα ∼ T (hα+1) · 1aα
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
the rough plan
is to recursively construct a mod fin increasing sequence{dα : α ∈ ω1} ⊂ J together with functions{hα : α ∈ ω1 + 1} ⊂ C (N, 2) so that for α < β ≤ ω1
1 hβ · 1dα =∗ hα (hence hα · 1dα =∗ hα)
2 there is aα ⊂ dα+1 \ dα in J ,
3 the family{〈aα ∩ T (hα+1)−1(< 1
3) , aα ∩ T (hα+1)−1(> 23)〉 : α ∈ ω1}
is shaping up to have the Hausdorff-Luzin property
4 so that T (hω1) can not(???) exist by (3)
Condition (4) poses a big problem because it basically requires that
T (hω1) · 1aα ∼ T (hα+1) · 1aα
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
the rough plan
is to recursively construct a mod fin increasing sequence{dα : α ∈ ω1} ⊂ J together with functions{hα : α ∈ ω1 + 1} ⊂ C (N, 2) so that for α < β ≤ ω1
1 hβ · 1dα =∗ hα (hence hα · 1dα =∗ hα)
2 there is aα ⊂ dα+1 \ dα in J ,
3 the family{〈aα ∩ T (hα+1)−1(< 1
3) , aα ∩ T (hα+1)−1(> 23)〉 : α ∈ ω1}
is shaping up to have the Hausdorff-Luzin property
4 so that T (hω1) can not(???) exist by (3)
Condition (4) poses a big problem because it basically requires that
T (hω1) · 1aα ∼ T (hα+1) · 1aα
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
We solve that with an idea I got from [BK]
Definition
Say that a, c ∈ J is a T -orthogonal pair if for all f ∈ C ∗(N),
T (f · 1c) · 1a ∼ 0
Lemma (PFA)
For any a, c ∈ J , there is a T -orthogonal pair a1, c1 such thata1 ⊂ a, c1 ⊂ c and Jc1 = Jc .
so when we are choosing our dα’s we will ensure, by induction,that JN\dα = ω, and aα, cα+1 = N \ dα+1 is a T -orthogonal pair.
this will mean what we do after stage α+ 1 will not affect what weaccomplished up to α
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
We solve that with an idea I got from [BK]
Definition
Say that a, c ∈ J is a T -orthogonal pair if for all f ∈ C ∗(N),
T (f · 1c) · 1a ∼ 0
Lemma (PFA)
For any a, c ∈ J , there is a T -orthogonal pair a1, c1 such thata1 ⊂ a, c1 ⊂ c and Jc1 = Jc .
so when we are choosing our dα’s we will ensure, by induction,that JN\dα = ω, and aα, cα+1 = N \ dα+1 is a T -orthogonal pair.
this will mean what we do after stage α+ 1 will not affect what weaccomplished up to α
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
We solve that with an idea I got from [BK]
Definition
Say that a, c ∈ J is a T -orthogonal pair if for all f ∈ C ∗(N),
T (f · 1c) · 1a ∼ 0
Lemma (PFA)
For any a, c ∈ J , there is a T -orthogonal pair a1, c1 such thata1 ⊂ a, c1 ⊂ c and Jc1 = Jc .
so when we are choosing our dα’s we will ensure, by induction,that JN\dα = ω, and aα, cα+1 = N \ dα+1 is a T -orthogonal pair.
this will mean what we do after stage α+ 1 will not affect what weaccomplished up to α
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
In actual fact, once we have chosen H = {hα, dα : α ∈ ω1} weobtain hω1 by ensuring that
Fn(N, 2, ω1)H = {f ∈ Fn(N, 2, ω1) : (∃α) f =∗ hα � dα}
is ccc. So, given dα, hα, set cα = N \ dα,
Lemma
For any T -orthogonal pair aα, cα+1 of subsets of cα and any family{Yn : n ∈ ω} of Fn(N, 2, ω1)hα-names of subsets of N, there is anhα+1 and dα+1 = N \ cα+1, hα+1 · 1dα = hα, andfor each n, Fn(N, 2, ω1)hα+1 forces that Yn does not separate(aα ∩ T (hα+1)−1(< 1
3)) and (aα ∩ T (hα+1)−1(> 23))
this creates a freezable gap, and then finish by freezing the gap
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
In actual fact, once we have chosen H = {hα, dα : α ∈ ω1} weobtain hω1 by ensuring that
Fn(N, 2, ω1)H = {f ∈ Fn(N, 2, ω1) : (∃α) f =∗ hα � dα}
is ccc. So, given dα, hα, set cα = N \ dα,
Lemma
For any T -orthogonal pair aα, cα+1 of subsets of cα and any family{Yn : n ∈ ω} of Fn(N, 2, ω1)hα-names of subsets of N, there is anhα+1 and dα+1 = N \ cα+1, hα+1 · 1dα = hα, andfor each n, Fn(N, 2, ω1)hα+1 forces that Yn does not separate(aα ∩ T (hα+1)−1(< 1
3)) and (aα ∩ T (hα+1)−1(> 23))
this creates a freezable gap, and then finish by freezing the gap
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?
In actual fact, once we have chosen H = {hα, dα : α ∈ ω1} weobtain hω1 by ensuring that
Fn(N, 2, ω1)H = {f ∈ Fn(N, 2, ω1) : (∃α) f =∗ hα � dα}
is ccc. So, given dα, hα, set cα = N \ dα,
Lemma
For any T -orthogonal pair aα, cα+1 of subsets of cα and any family{Yn : n ∈ ω} of Fn(N, 2, ω1)hα-names of subsets of N, there is anhα+1 and dα+1 = N \ cα+1, hα+1 · 1dα = hα, andfor each n, Fn(N, 2, ω1)hα+1 forces that Yn does not separate(aα ∩ T (hα+1)−1(< 1
3)) and (aα ∩ T (hα+1)−1(> 23))
this creates a freezable gap, and then finish by freezing the gap
Alan Dow Is `∞/c0 i.e. ((C(N∗), ‖ · ‖)) primary?