There is no bound on sizes of indecomposable Banach spaces l - - PowerPoint PPT Presentation

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There is no bound on sizes of indecomposable Banach spaces l - - PowerPoint PPT Presentation

Oct 12, 2022 276 likes •740 views

There is no bound on sizes of indecomposable Banach spaces l Micha Swi etek Faculty of Mathematics and Computer Science Jagiellonian University Joint work with Piotr Koszmider and Saharon Shelah B edlewo 2016 l Micha

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There is no bound on sizes of indecomposable Banach spaces

Micha l ´ Swi¸ etek

Faculty of Mathematics and Computer Science Jagiellonian University

Joint work with Piotr Koszmider and Saharon Shelah

B¸ edlewo 2016

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 1 / 12

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The old question

The old question

Is every infinite dimensional Banach space decomposable? (i.e. it can be decomposed as a direct sum of two infinite dimensional closed subspaces)

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 2 / 12

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The old question

The old question

Is every infinite dimensional Banach space decomposable? (i.e. it can be decomposed as a direct sum of two infinite dimensional closed subspaces) yes for ancients: c0, ℓp, Lp, C([0, 1])

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 2 / 12

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The old question

The old question

Is every infinite dimensional Banach space decomposable? (i.e. it can be decomposed as a direct sum of two infinite dimensional closed subspaces) yes for ancients: c0, ℓp, Lp, C([0, 1]) yes for nonseparable WCG spaces (Amir, Lindenstrauss)

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 2 / 12

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The old question

The old question

Is every infinite dimensional Banach space decomposable? (i.e. it can be decomposed as a direct sum of two infinite dimensional closed subspaces) yes for ancients: c0, ℓp, Lp, C([0, 1]) yes for nonseparable WCG spaces (Amir, Lindenstrauss) yes for duals of nonseparable Banach spaces (Heinrich, Mankiewicz)

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 2 / 12

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The old question

The old question

Is every infinite dimensional Banach space decomposable? (i.e. it can be decomposed as a direct sum of two infinite dimensional closed subspaces) yes for ancients: c0, ℓp, Lp, C([0, 1]) yes for nonseparable WCG spaces (Amir, Lindenstrauss) yes for duals of nonseparable Banach spaces (Heinrich, Mankiewicz) no in general: Gowers, Maurey (separable) and Argyros (of density continuum)

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 2 / 12

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The old question

The old question

Is every infinite dimensional Banach space decomposable? (i.e. it can be decomposed as a direct sum of two infinite dimensional closed subspaces) yes for ancients: c0, ℓp, Lp, C([0, 1]) yes for nonseparable WCG spaces (Amir, Lindenstrauss) yes for duals of nonseparable Banach spaces (Heinrich, Mankiewicz) no in general: Gowers, Maurey (separable) and Argyros (of density continuum) the constructed spaces have even stronger property - they are hereditarily indecomposable (every closed subspace is not decomposable)

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 2 / 12

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How big indecomposable space can be?

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 3 / 12

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How big indecomposable space can be?

Theorem (Plichko, Yost ’00)

If a Banach space X does not admit an injective operator into ℓ∞, in particular if its density character exeeds the continuum, then X has a decomposable subspace.

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 3 / 12

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How big indecomposable space can be?

Theorem (Plichko, Yost ’00)

If a Banach space X does not admit an injective operator into ℓ∞, in particular if its density character exeeds the continuum, then X has a decomposable subspace.

A problem of Argyros

Is there a bound on densities of indecomposable Banach spaces?

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 3 / 12

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How big indecomposable space can be?

Theorem (Plichko, Yost ’00)

If a Banach space X does not admit an injective operator into ℓ∞, in particular if its density character exeeds the continuum, then X has a decomposable subspace.

A problem of Argyros

Is there a bound on densities of indecomposable Banach spaces?

New indecomposables!

Koszmider ’04: indecomposable Banach space of density continuum of the form C(K) (under CH).

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 3 / 12

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Basic definitions

Definition

Let K be a compact topological spaces:

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 4 / 12

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Basic definitions

Definition

Let K be a compact topological spaces: we say that C(K) has few operators, if every operator T on C(K) is

  • f the form T = Mg + W , where g ∈ C(K), Mg is a multiplication
  • perator, and W is a weakly compact operator,

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 4 / 12

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Basic definitions

Definition

Let K be a compact topological spaces: we say that C(K) has few operators, if every operator T on C(K) is

  • f the form T = Mg + W , where g ∈ C(K), Mg is a multiplication
  • perator, and W is a weakly compact operator,

we say that C(K) has few∗ operators, if for every operator T on C(K) the operator T ∗ is of the form T ∗ = M∗

g + W , where

g : K → R is a bounded borel function, and W is a weakly compact

  • perator,

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 4 / 12

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Basic definitions

Definition

Let K be a compact topological spaces: we say that C(K) has few operators, if every operator T on C(K) is

  • f the form T = Mg + W , where g ∈ C(K), Mg is a multiplication
  • perator, and W is a weakly compact operator,

we say that C(K) has few∗ operators, if for every operator T on C(K) the operator T ∗ is of the form T ∗ = M∗

g + W , where

g : K → R is a bounded borel function, and W is a weakly compact

  • perator,

we say that x ∈ K is a butterfly point, if there are disjoint open sets U, V ⊂ K such that {x} = U ∩ V .

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 4 / 12

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Theorem (Koszmider ’04)

If K is a compact connected space and C(K) has few operators, then C(K) is indecomposable.

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 5 / 12

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Theorem (Koszmider ’04)

If K is a compact connected space and C(K) has few operators, then C(K) is indecomposable.

Theorem (Koszmider ’04)

If K is a compact space without butterfly points and C(K) has few*

  • perators, then it has in fact few operators.

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 5 / 12

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Example

Among others there are constructions of compact sets K with the following properties

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 6 / 12

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Example

Among others there are constructions of compact sets K with the following properties ’04 Koszmider: K totally disconnected, C(K) with few* operators,

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 6 / 12

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Example

Among others there are constructions of compact sets K with the following properties ’04 Koszmider: K totally disconnected, C(K) with few* operators, ’04 Koszmider: K connected, C(K) few operators (under CH),

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 6 / 12

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Example

Among others there are constructions of compact sets K with the following properties ’04 Koszmider: K totally disconnected, C(K) with few* operators, ’04 Koszmider: K connected, C(K) few operators (under CH), ’04 Plebanek: K connected, C(K) few operators,

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 6 / 12

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Example

Among others there are constructions of compact sets K with the following properties ’04 Koszmider: K totally disconnected, C(K) with few* operators, ’04 Koszmider: K connected, C(K) few operators (under CH), ’04 Plebanek: K connected, C(K) few operators, ’13 Koszmider: K connected, C(K) few operators of density 22ℵ0 (under ℵ1 = 2ℵ0, ℵ2 = 2ℵ1).

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 6 / 12

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Theorem (Koszmider, Shelah, ´ S.)

Assume the generalized continuum hypothesis. For every cardinal λ there is an indecomposable Banach space of density bigger than λ.

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 7 / 12

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Theorem (Koszmider, Shelah, ´ S.)

Assume the generalized continuum hypothesis. For every cardinal λ there is an indecomposable Banach space of density bigger than λ. Our strategy of the proof:

Theorem

If K is a compact connected space without butterfly points and C(K) has few* operators, then the space C(K) is indecomposable.

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 7 / 12

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Theorem (Koszmider, Shelah, ´ S.)

Assume the generalized continuum hypothesis. For every cardinal λ there is an indecomposable Banach space of density bigger than λ. Our strategy of the proof:

Theorem

If K is a compact connected space without butterfly points and C(K) has few* operators, then the space C(K) is indecomposable.

1 K is large (of weight greater than λ), 2 K is compact, 3 K has no butterfly points, 4 K is connected, 5 C(K) has few* operators. Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 7 / 12

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Definitions and basic facts

We fix a regular cardinal κ > λ such that κω = κ,

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 8 / 12

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Definitions and basic facts

We fix a regular cardinal κ > λ such that κω = κ, L = S(Fr(κ)),

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 8 / 12

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Definitions and basic facts

We fix a regular cardinal κ > λ such that κω = κ, L = S(Fr(κ)), (L ” = ” [0, 1]κ)

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 8 / 12

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Definitions and basic facts

We fix a regular cardinal κ > λ such that κω = κ, L = S(Fr(κ)), (L ” = ” [0, 1]κ) D = {dα | α < κ} - ”coordinate system”,

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 8 / 12

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Definitions and basic facts

We fix a regular cardinal κ > λ such that κω = κ, L = S(Fr(κ)), (L ” = ” [0, 1]κ) D = {dα | α < κ} - ”coordinate system”, F ⊂ CI(L). Let ΠF : L → [0, 1]F, ΠF(x)(f ) = f (x), x ∈ L, f ∈ F,

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 8 / 12

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Definitions and basic facts

We fix a regular cardinal κ > λ such that κω = κ, L = S(Fr(κ)), (L ” = ” [0, 1]κ) D = {dα | α < κ} - ”coordinate system”, F ⊂ CI(L). Let ΠF : L → [0, 1]F, ΠF(x)(f ) = f (x), x ∈ L, f ∈ F, ∇F = ΠF[L],

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 8 / 12

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Definitions and basic facts

We fix a regular cardinal κ > λ such that κω = κ, L = S(Fr(κ)), (L ” = ” [0, 1]κ) D = {dα | α < κ} - ”coordinate system”, F ⊂ CI(L). Let ΠF : L → [0, 1]F, ΠF(x)(f ) = f (x), x ∈ L, f ∈ F, ∇F = ΠF[L], if F does not depends on α, then ∇(F ∪ {dα}) = ∇F × [0, 1],

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 8 / 12

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Definitions and basic facts

We fix a regular cardinal κ > λ such that κω = κ, L = S(Fr(κ)), (L ” = ” [0, 1]κ) D = {dα | α < κ} - ”coordinate system”, F ⊂ CI(L). Let ΠF : L → [0, 1]F, ΠF(x)(f ) = f (x), x ∈ L, f ∈ F, ∇F = ΠF[L], if F does not depends on α, then ∇(F ∪ {dα}) = ∇F × [0, 1], ∇D = [0, 1]κ,

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 8 / 12

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Definitions and basic facts

We fix a regular cardinal κ > λ such that κω = κ, L = S(Fr(κ)), (L ” = ” [0, 1]κ) D = {dα | α < κ} - ”coordinate system”, F ⊂ CI(L). Let ΠF : L → [0, 1]F, ΠF(x)(f ) = f (x), x ∈ L, f ∈ F, ∇F = ΠF[L], if F does not depends on α, then ∇(F ∪ {dα}) = ∇F × [0, 1], ∇D = [0, 1]κ, for every D ⊂ F ⊂ CI(L) the space ∇F satisfies (1) largeness, (2) compactness, (3) no butterflies,

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 8 / 12

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Definitions and basic facts

We fix a regular cardinal κ > λ such that κω = κ, L = S(Fr(κ)), (L ” = ” [0, 1]κ) D = {dα | α < κ} - ”coordinate system”, F ⊂ CI(L). Let ΠF : L → [0, 1]F, ΠF(x)(f ) = f (x), x ∈ L, f ∈ F, ∇F = ΠF[L], if F does not depends on α, then ∇(F ∪ {dα}) = ∇F × [0, 1], ∇D = [0, 1]κ, for every D ⊂ F ⊂ CI(L) the space ∇F satisfies (1) largeness, (2) compactness, (3) no butterflies, F = D ∪ {gα | α ∈ E κ

ω}.

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 8 / 12

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few*

K a compact set, T operator on C(K) such that T ∗ is not of the form M∗

g + W .

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 9 / 12

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few*

K a compact set, T operator on C(K) such that T ∗ is not of the form M∗

g + W .

Then, there are ε > 0, (fn) ⊂ CI(K) an antichain (fnfm = 0) and (Un) ⊂ top(K) an antichain (Un ∩ Um = ∅) such that supp(fn) ∩ Um = ∅ and |T(fn)| ↾ Un > ε

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 9 / 12

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few*

K a compact set, T operator on C(K) such that T ∗ is not of the form M∗

g + W .

Then, there are ε > 0, (fn) ⊂ CI(K) an antichain (fnfm = 0) and (Un) ⊂ top(K) an antichain (Un ∩ Um = ∅) such that supp(fn) ∩ Um = ∅ and |T(fn)| ↾ Un > ε Doing some combinatorics we can find an infinite set M ⊂ N such that for a supremum f of a sequence (fn)n∈M we have |T(f )| ↾ Un > 2ε/3, n ∈ M and |T(f )| ↾ Un < ε/3, n ∈ M,

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 9 / 12

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few*

K a compact set, T operator on C(K) such that T ∗ is not of the form M∗

g + W .

Then, there are ε > 0, (fn) ⊂ CI(K) an antichain (fnfm = 0) and (Un) ⊂ top(K) an antichain (Un ∩ Um = ∅) such that supp(fn) ∩ Um = ∅ and |T(fn)| ↾ Un > ε Doing some combinatorics we can find an infinite set M ⊂ N such that for a supremum f of a sequence (fn)n∈M we have |T(f )| ↾ Un > 2ε/3, n ∈ M and |T(f )| ↾ Un < ε/3, n ∈ M, which gives that

n∈M Un ∩ n∈M Un = ∅.

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 9 / 12

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Assymptotic distribution of separations

Let κ be a cardinal, K be a compact Hausdorff space and let dα : K → [0, 1] be continuous for every α < κ. Let dα,1 = dα and dα,−1 = 1 − dα. C(K) is said to have asymmetric distribution of separations in the direction of D = (dα | α < κ) if and only if Given (i) (fn)n∈N ⊂ C(K) an antichain, (ii) (Un)n∈N ⊂ top(K) an antichain s.t. supp(fn) ∩ Um = ∅ for n, m ∈ N, (iii) (νξ

n)n∈N ⊆ {−1, 1} for all ξ ∈ κ,

(iv) { (Uξ

n)n∈N | ξ ∈ κ } ⊂ top(K) such that Uξ n ⊆ Un for n ∈ N, ξ ∈ κ;

There exist an increasing sequence (ηn)n∈N ⊂ κ and an infinite, coinfinite M ⊂ N such that (a) the supremum

n∈M

  • fn · dηn,νηn

n

  • exists in C(K),

(b)

n∈M Uηn n ∩ n∈N\M Uηn n = ∅.

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 10 / 12

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Theorem (Koszmider, Shelah, ´ S.)

Assume the generalized continuum hypothesis. Let κ be the successor of a cardinal of uncountable cofinality. There is a compact Hausdorff connected c.c.c. space K of weight κ without a butterfly point such that C(K) has asymmetric distribution of separations in the direction of some D ⊂ CI(K).

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 11 / 12

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Theorem (Koszmider, Shelah, ´ S.)

Assume the generalized continuum hypothesis. Let κ be the successor of a cardinal of uncountable cofinality. There is a compact Hausdorff connected c.c.c. space K of weight κ without a butterfly point such that C(K) has asymmetric distribution of separations in the direction of some D ⊂ CI(K). This gives few* operators, but how about connectedness?

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 11 / 12

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Theorem (Koszmider, Shelah, ´ S.)

Assume the generalized continuum hypothesis. Let κ be the successor of a cardinal of uncountable cofinality. There is a compact Hausdorff connected c.c.c. space K of weight κ without a butterfly point such that C(K) has asymmetric distribution of separations in the direction of some D ⊂ CI(K). This gives few* operators, but how about connectedness? If α ∈ E κ

ω, then we take some antichain (fn) on ∇(F<α) and add a

function gα from C(L) such that the antichain (fn)n∈M, for some M, has a supremum in C(∇(F<α ∪ {dα, gα})).

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 11 / 12

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Theorem (Koszmider, Shelah, ´ S.)

Assume the generalized continuum hypothesis. Let κ be the successor of a cardinal of uncountable cofinality. There is a compact Hausdorff connected c.c.c. space K of weight κ without a butterfly point such that C(K) has asymmetric distribution of separations in the direction of some D ⊂ CI(K). This gives few* operators, but how about connectedness? If α ∈ E κ

ω, then we take some antichain (fn) on ∇(F<α) and add a

function gα from C(L) such that the antichain (fn)n∈M, for some M, has a supremum in C(∇(F<α ∪ {dα, gα})). Then we have that ∇(F<α ∪ {dα, gα}) = Γ(

  • n∈M

fn).

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 11 / 12

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THANK YOU FOR YOUR ATTENTION

Micha l ´ Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 12 / 12