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� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 1 / 12
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) l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 2 / 12
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: c 0 , ℓ p , L p , C ([0 , 1]) l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 2 / 12
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: c 0 , ℓ p , L p , C ([0 , 1]) yes for nonseparable WCG spaces (Amir, Lindenstrauss) l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 2 / 12
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: c 0 , ℓ p , L p , C ([0 , 1]) yes for nonseparable WCG spaces (Amir, Lindenstrauss) yes for duals of nonseparable Banach spaces (Heinrich, Mankiewicz) l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 2 / 12
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: c 0 , ℓ p , L p , 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) l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 2 / 12
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: c 0 , ℓ p , L p , 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) l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 2 / 12
How big indecomposable space can be? l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 3 / 12
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. l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 3 / 12
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? l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 3 / 12
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). l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 3 / 12
Basic definitions Definition Let K be a compact topological spaces: l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 4 / 12
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 of the form T = M g + W , where g ∈ C ( K ), M g is a multiplication operator, and W is a weakly compact operator, l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 4 / 12
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 of the form T = M g + W , where g ∈ C ( K ), M g is a multiplication operator, 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 operator, l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 4 / 12
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 of the form T = M g + W , where g ∈ C ( K ), M g is a multiplication operator, 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 operator, we say that x ∈ K is a butterfly point , if there are disjoint open sets U , V ⊂ K such that { x } = U ∩ V . l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 4 / 12
Theorem (Koszmider ’04) If K is a compact connected space and C ( K ) has few operators, then C ( K ) is indecomposable. l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 5 / 12
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* operators, then it has in fact few operators. l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 5 / 12
Example Among others there are constructions of compact sets K with the following properties l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 6 / 12
Example Among others there are constructions of compact sets K with the following properties ’04 Koszmider: K totally disconnected, C ( K ) with few* operators, l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 6 / 12
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), l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 6 / 12
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, l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 6 / 12
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 2 2 ℵ 0 (under ℵ 1 = 2 ℵ 0 , ℵ 2 = 2 ℵ 1 ). l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 6 / 12
Theorem (Koszmider, Shelah, ´ S.) Assume the generalized continuum hypothesis. For every cardinal λ there is an indecomposable Banach space of density bigger than λ . l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 7 / 12
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. l ´ Micha� Swi¸ etek (Jagiellonian University) There in no bound on sizes ... B¸ edlewo 2016 7 / 12
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