The elusive geometry of the Banach space `

(1)

The elusive geometry of the Banach space `

_∞

/c

₀

Piotr Koszmider

IM PAN, Warsaw

(2)

Banach spaces and topological preliminaries

1 `∞={(a_n)n∈N:a_n∈R,(a_n)n∈Nbounded} ≡C(βN), sup norm

2 c₀={(a_n)n∈N∈`∞:limn→∞a_n =0} ≡

3 ≡ {f ∈C(βN) :f (βN\N) =0}, sup norm

4 N^∗ =βN\N

5 `∞/c₀≡C(N^∗)

6 Clop(βN)≡℘(N)

7 Clop(N^∗)≡℘(N)/Fin

(3)

Banach spaces and topological preliminaries

2 c₀={(a_n)n∈N∈`∞:limn→∞a_n =0} ≡

4 N^∗ =βN\N

5 `∞/c₀≡C(N^∗)

(4)

Banach spaces and topological preliminaries

2 c₀={(a_n)n∈N∈`∞:limn→∞a_n =0} ≡

4 N^∗ =βN\N

5 `∞/c₀≡C(N^∗)

(5)

Banach spaces and topological preliminaries

2 c₀={(a_n)n∈N∈`∞:limn→∞a_n =0} ≡

4 N^∗ =βN\N

5 `∞/c₀≡C(N^∗)

(6)

Banach spaces and topological preliminaries

2 c₀={(a_n)n∈N∈`∞:limn→∞a_n =0} ≡

4 N^∗ =βN\N

5 `∞/c₀≡C(N^∗)

(7)

Banach spaces and topological preliminaries

2 c₀={(a_n)n∈N∈`∞:limn→∞a_n =0} ≡

4 N^∗ =βN\N

5 `∞/c₀≡C(N^∗)

(8)

Banach spaces and topological preliminaries

2 c₀={(a_n)n∈N∈`∞:limn→∞a_n =0} ≡

4 N^∗ =βN\N

5 `∞/c₀≡C(N^∗)

(9)

Banach spaces and topological preliminaries

2 c₀={(a_n)n∈N∈`∞:limn→∞a_n =0} ≡

4 N^∗ =βN\N

5 `∞/c₀≡C(N^∗)

(10)

Banach spaces and topological preliminaries

2 c₀={(a_n)n∈N∈`∞:limn→∞a_n =0} ≡

4 N^∗ =βN\N

5 `∞/c₀≡C(N^∗)

(11)

Banach spaces and topological preliminaries

2 c₀={(a_n)n∈N∈`∞:limn→∞a_n =0} ≡

4 N^∗ =βN\N

5 `∞/c₀≡C(N^∗)

(12)

Set-theoretic and logical preliminaries

1 CH = ”All infinite subsets ofRhave the cardinality ofRor the cardinality ofN”,

2 There is a well ordering of`∞/c₀which has all proper initial segments countable,

3 (2) is useful for transfinite inductive construction of length

|`_∞/c₀|=|℘(N)|=|R|=ω1

4 Alternative axioms: MA+not CH, OCA, PFA.

5 Vocabulary: in ZFC, consistent, cannot be proved

(13)

Set-theoretic and logical preliminaries

|`_∞/c₀|=|℘(N)|=|R|=ω1

(14)

Set-theoretic and logical preliminaries

|`_∞/c₀|=|℘(N)|=|R|=ω1

(15)

Set-theoretic and logical preliminaries

|`_∞/c₀|=|℘(N)|=|R|=ω1

(16)

Set-theoretic and logical preliminaries

|`_∞/c₀|=|℘(N)|=|R|=ω1

(17)

Set-theoretic and logical preliminaries

|`_∞/c₀|=|℘(N)|=|R|=ω1

(18)

Set-theoretic and logical preliminaries

|`_∞/c₀|=|℘(N)|=|R|=ω1

(19)

Set-theoretic and logical preliminaries

|`_∞/c₀|=|℘(N)|=|R|=ω1

(20)

Outline of the talk

1 The universality of`∞/c₀

2 Complemented subspaces of`∞/c₀

3 Complemented copies of`∞/c₀in`∞/c₀

4 Infinite decompositions of`∞/c₀

5 The primaryness of`∞/c₀

6 Automorphisms of`∞/c₀

(21)

Outline of the talk

(22)

Outline of the talk

(23)

Outline of the talk

(24)

Outline of the talk

4 Infinite decompositions of`∞/c0

(25)

Outline of the talk

(26)

Outline of the talk

(27)

Outline of the talk

(28)

Outline of the talk

(29)

Universality of `

_∞

/c

₀

1 (CH) [Esenin-Volpin, Doklady 1949] For any Banach spaceX of density continuum, there is an isometric embedding ofX into

`∞/c₀

2 Spaces of density 2^ω which can be embedded into`∞/c₀without using CH:

1 all Banach spaces of densityω₁,

2 c0(2^ω), many otherC(K)s

3 `_p(2^ω)for 1≤p<∞,

3 (Brech, P.K. - IJM 2012) It is consistent thatC([0,2^ω])does not embed isomorphically into`∞/c₀

4 (Brech, P.K. - IJM 2012) It is consistent that there is no isomorphically universal Banach space of density 2^ω

5 Some WCG or Hilbert generatedC(K)s consistently do not embed in`∞/c₀(Todorcevic - JMAA 2012, Krupski, Marciszewski - Coll.M 2012, Brech, P.K. - PAMS 2013)

(30)

Universality of `

_∞

/c

₀

`∞/c₀

3 `_p(2^ω)for 1≤p<∞,

(31)

Universality of `

_∞

/c

₀

`∞/c₀

3 `_p(2^ω)for 1≤p<∞,

(32)

Universality of `

_∞

/c

₀

`∞/c₀

3 `_p(2^ω)for 1≤p<∞,

(33)

Universality of `

_∞

/c

₀

`∞/c₀

3 `_p(2^ω)for 1≤p<∞,

(34)

Universality of `

_∞

/c

₀

`∞/c₀

3 `_p(2^ω)for 1≤p<∞,

(35)

Universality of `

_∞

/c

₀

`∞/c₀

3 `_p(2^ω)for 1≤p<∞,

(36)

Universality of `

_∞

/c

₀

`∞/c₀

3 `_p(2^ω)for 1≤p<∞,

4 (Brech, P.K. - IJM 2012) It is consistent that there is no

(37)

Universality of `

_∞

/c

₀

`∞/c₀

3 `_p(2^ω)for 1≤p<∞,

(38)

Universality of `

_∞

/c

₀

`∞/c₀

3 `_p(2^ω)for 1≤p<∞,

(39)

Universality of `

_∞

/c

₀

`∞/c₀

3 `_p(2^ω)for 1≤p<∞,

(40)

Complemented subspaces of `

_∞

/c

₀

1 `∞/c₀≡(`∞/c₀)⊕(`∞/c₀)

2 `∞/c₀≡`∞⊕(`∞/c₀)

3 (CH)`∞/c₀≡L∞([0,1]^ω¹)⊕(`∞/c₀)

Question

Does L∞([0,1]^ω¹)embed into`∞/c₀in ZFC?

(41)

Complemented subspaces of `

_∞

/c

₀

1 `∞/c₀≡(`∞/c₀)⊕(`∞/c₀)

2 `∞/c₀≡`∞⊕(`∞/c₀)

3 (CH)`∞/c₀≡L∞([0,1]^ω¹)⊕(`∞/c₀) Question

(42)

Complemented subspaces of `

_∞

/c

₀

1 `∞/c₀≡(`∞/c₀)⊕(`∞/c₀)

2 `∞/c₀≡`∞⊕(`∞/c₀)

3 (CH)`∞/c₀≡L∞([0,1]^ω¹)⊕(`∞/c₀) Question

(43)

Complemented subspaces of `

_∞

/c

₀

1 `∞/c₀≡(`∞/c₀)⊕(`∞/c₀)

2 `∞/c₀≡`∞⊕(`∞/c₀)

3 (CH)`∞/c₀≡L∞([0,1]^ω¹)⊕(`∞/c₀)

Question

(44)

Complemented subspaces of `

_∞

/c

₀

1 `∞/c₀≡(`∞/c₀)⊕(`∞/c₀)

2 `∞/c₀≡`∞⊕(`∞/c₀)

3 (CH)`∞/c₀≡L∞([0,1]^ω¹)⊕(`∞/c₀)

Question

(45)

Complemented subspaces of `

_∞

/c

₀

1 `∞/c₀≡(`∞/c₀)⊕(`∞/c₀)

2 `∞/c₀≡`∞⊕(`∞/c₀)

3 (CH)`∞/c₀≡L∞([0,1]^ω¹)⊕(`∞/c₀) Question

(46)

Complemented subspaces of `

_∞

/c

₀

1 `∞/c₀≡(`∞/c₀)⊕(`∞/c₀)

2 `∞/c₀≡`∞⊕(`∞/c₀)

3 (CH)`∞/c₀≡L∞([0,1]^ω¹)⊕(`∞/c₀) Question

(47)

Complemented copies of `

_∞

/c

₀

Theorem (Castillo, Plichko; JFA 2010)

(CH) There are uncomplemented subspaces of`∞/c₀isomorphic to

`∞/c₀.

Problem

Does every subspace of`∞/c₀isomorphic to`∞/c₀contains a further subspace isomorphic to`∞/c₀which is complemented in the entire

`∞/c₀?

Theorem (Drewnowski, Roberts - PAMS 1991) Whenever`∞/c₀=A⊕B, then either A or B contains a complemented copy of`∞/c₀.

(48)

Complemented copies of `

_∞

/c

₀

`∞/c₀.

Problem

`∞/c₀?

(49)

Complemented copies of `

_∞

/c

₀

`∞/c₀.

Problem

`∞/c₀?

(50)

Complemented copies of `

_∞

/c

₀

`∞/c₀.

Problem

`∞/c₀?

Theorem (Drewnowski, Roberts - PAMS 1991)

(51)

Complemented copies of `

_∞

/c

₀

`∞/c₀.

Problem

`∞/c₀?

(52)

Pełczy ´nski decomposition method (Studia M. 1960)

Theorem

Suppose that X,Y are Banach spaces and

1 X is isomorphic to a complemented subspace of Y ,

2 Y is isomorphic to a complemented subspace of X ,

3 X is isomorphic to`∞(X), Then X and Y are isomorphic.

(53)

Pełczy ´nski decomposition method (Studia M. 1960)

Theorem

Suppose that X,Y are Banach spaces and

1 X is isomorphic to a complemented subspace of Y ,

2 Y is isomorphic to a complemented subspace of X ,

3 X is isomorphic to`∞(X), Then X and Y are isomorphic.

(54)

`∞-sums

Theorem (Negrepontis - TAMS 1969) (CH)

`∞/c₀∼`∞(`∞/c₀).

Theorem (Brech, P.K. - Fund. M. 2014)

It is consistent that`∞/c₀is not isomorphic to any Banach space of the form`∞(X)

(55)

`∞-sums

`∞/c₀∼`∞(`∞/c₀).

(56)

`∞-sums

`∞/c₀∼`∞(`∞/c₀).

(57)

`∞-sums

`∞/c₀∼`∞(`∞/c₀).

(58)

The primariness of `

_∞

/c

₀

(CH) Whenever`∞/c₀=X⊕Y , then either X or Y is isomorphic to

`∞/c₀.

Question

Is`∞/c₀primary in ZFC?

Theorem (Dow, Shelah - Top. Ap. 2008)

It is consistent that there are two disjoint open subsets U,V ⊆N^∗ with U∩V ={x}for some x ∈N^∗ and neither U nor V is homeomorphic to N^∗.

(59)

The primariness of `

_∞

/c

₀

(CH) Whenever`∞/c₀=X ⊕Y , then either X or Y is isomorphic to

`∞/c₀.

Question

(60)

The primariness of `

_∞

/c

₀

`∞/c₀.

Question

(61)

The primariness of `

_∞

/c

₀

`∞/c₀.

Question

It is consistent that there are two disjoint open subsets U,V ⊆N^∗ with U∩V ={x}for some x ∈N^∗ and neither U nor V is homeomorphic to

(62)

Automorphisms

Joint work in preparation with Crist ´obal Rodriguez Porras

1 There is an an automorphism of`∞/c₀which is not induced by an operator on`∞

2 We develop local canonization of some automorphisms under OCA+MA

(63)

Automorphisms

(64)

Automorphisms

(65)

Automorphisms

(66)

Automorphisms

(67)

Automorphisms

(68)

Some references:

L. Drewnowski, J. Roberts,On the primariness of the Banach space

`∞/c0. Proc. Amer. Math. Soc. 112 (1991), no. 4, 949–957.

C. Brech, P. Koszmider,On universal Banach spaces of density continuum. Israel J. Math. 190 (2012), 93–110.

C. Brech, P. Koszmider,`∞-sums and the Banach space`∞/c₀. Fund.

Math. 224 (2014), 175–185.

P. Koszmider, C. Rodriguez Porras; in preparation.