A Chain condition for operators from C(K)-spaces

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  1. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesA chain condition for operators from C (K )-spaces Quidquid latine…
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  • 1. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesA chain condition for operators from C (K )-spaces Quidquid latine dictum sit, altum videtur K. P. Hart Faculty EEMCS TU Delft Warszawa, 19 kwietnia, 2013: 09:00 – 10:05 K. P. Hart A chain condition for operators from C (K )-spaces
  • 2. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesOutline 1 Weakly compact operators 2 A chain condition 3 Spaces with and without uncountable δ -chains 4 Sources K. P. Hart A chain condition for operators from C (K )-spaces
  • 3. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesOutline 1 Weakly compact operators 2 A chain condition 3 Spaces with and without uncountable δ -chains 4 Sources K. P. Hart A chain condition for operators from C (K )-spaces
  • 4. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesPelczy´ski’s Theorem n Confusingly (for a topologist): K generally denotes a compact space, X generally denotes a Banach space. K. P. Hart A chain condition for operators from C (K )-spaces
  • 5. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesPelczy´ski’s Theorem n Confusingly (for a topologist): K generally denotes a compact space, X generally denotes a Banach space. Theorem An operator T : C (K ) → X is weakly compact iff there is no isomorphic copy of c0 on which T is invertible. K. P. Hart A chain condition for operators from C (K )-spaces
  • 6. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesReformulation An operator T : C (K ) → X is not weakly compact iff there is a sequence fn : n < ω of continuous functions such that K. P. Hart A chain condition for operators from C (K )-spaces
  • 7. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesReformulation An operator T : C (K ) → X is not weakly compact iff there is a sequence fn : n < ω of continuous functions such that fn 1 for all n K. P. Hart A chain condition for operators from C (K )-spaces
  • 8. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesReformulation An operator T : C (K ) → X is not weakly compact iff there is a sequence fn : n < ω of continuous functions such that fn 1 for all n supp fm ∩ supp fn = ∅ whenever m = n K. P. Hart A chain condition for operators from C (K )-spaces
  • 9. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesReformulation An operator T : C (K ) → X is not weakly compact iff there is a sequence fn : n < ω of continuous functions such that fn 1 for all n supp fm ∩ supp fn = ∅ whenever m = n inf n Tfn > 0 K. P. Hart A chain condition for operators from C (K )-spaces
  • 10. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesOutline 1 Weakly compact operators 2 A chain condition 3 Spaces with and without uncountable δ -chains 4 Sources K. P. Hart A chain condition for operators from C (K )-spaces
  • 11. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesWhere’s the chain? First: an order on C (K ). K. P. Hart A chain condition for operators from C (K )-spaces
  • 12. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesWhere’s the chain? First: an order on C (K ). We say f g if K. P. Hart A chain condition for operators from C (K )-spaces
  • 13. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesWhere’s the chain? First: an order on C (K ). We say f g if f =g K. P. Hart A chain condition for operators from C (K )-spaces
  • 14. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesWhere’s the chain? First: an order on C (K ). We say f g if f =g g supp f = f supp f K. P. Hart A chain condition for operators from C (K )-spaces
  • 15. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesWhere’s the chain? First: an order on C (K ). We say f g if f =g g supp f = f supp f Second: another order on C (K ). K. P. Hart A chain condition for operators from C (K )-spaces
  • 16. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesWhere’s the chain? First: an order on C (K ). We say f g if f =g g supp f = f supp f Second: another order on C (K ). Let δ > 0; we say f δ g if K. P. Hart A chain condition for operators from C (K )-spaces
  • 17. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesWhere’s the chain? First: an order on C (K ). We say f g if f =g g supp f = f supp f Second: another order on C (K ). Let δ > 0; we say f δ g if g −f δ K. P. Hart A chain condition for operators from C (K )-spaces
  • 18. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesWhere’s the chain? First: an order on C (K ). We say f g if f =g g supp f = f supp f Second: another order on C (K ). Let δ > 0; we say f δ g if g −f δ g supp f = f supp f K. P. Hart A chain condition for operators from C (K )-spaces
  • 19. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesWhere’s the chain? First: an order on C (K ). We say f g if f =g g supp f = f supp f Second: another order on C (K ). Let δ > 0; we say f δ g if g −f δ g supp f = f supp f The speaker draws an instructive picture. K. P. Hart A chain condition for operators from C (K )-spaces
  • 20. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesHere’s the chain An operator T : C (K ) → X is not weakly compact iff there is an infinite -chain, C , such that K. P. Hart A chain condition for operators from C (K )-spaces
  • 21. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesHere’s the chain An operator T : C (K ) → X is not weakly compact iff there is an infinite -chain, C , such that inf Tf − Tg : {f , g } ∈ [C ]2 > 0 K. P. Hart A chain condition for operators from C (K )-spaces
  • 22. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesHere’s the chain An operator T : C (K ) → X is not weakly compact iff there is an infinite -chain, C , such that inf Tf − Tg : {f , g } ∈ [C ]2 > 0 Proof. Given fn : n < ω let gn = i n fi ; then gn : n < ω is a (bad) chain. K. P. Hart A chain condition for operators from C (K )-spaces
  • 23. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesHere’s the chain An operator T : C (K ) → X is not weakly compact iff there is an infinite -chain, C , such that inf Tf − Tg : {f , g } ∈ [C ]2 > 0 Proof. Given fn : n < ω let gn = i n fi ; then gn : n < ω is a (bad) chain. Given an infinite chain, C , take a monotone sequence gn : n < ω in C and let fn = gn+1 − gn for all n. K. P. Hart A chain condition for operators from C (K )-spaces
  • 24. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesHere is the chain condition B For every uncountable -chain in C (K ) we have inf f − g : {f , g } ∈ [C ]2 = 0 K. P. Hart A chain condition for operators from C (K )-spaces
  • 25. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesHere is the chain condition B For every uncountable -chain in C (K ) we have inf f − g : {f , g } ∈ [C ]2 = 0 In other words: K. P. Hart A chain condition for operators from C (K )-spaces
  • 26. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesHere is the chain condition B For every uncountable -chain in C (K ) we have inf f − g : {f , g } ∈ [C ]2 = 0 In other words: B For every δ > 0: every δ -chain is countable. K. P. Hart A chain condition for operators from C (K )-spaces
  • 27. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesWhy ‘uncountable’ ? Well, . . . K. P. Hart A chain condition for operators from C (K )-spaces
  • 28. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesWhy ‘uncountable’ ? Well, . . . Theorem If K is extremally disconnected then T : C (K ) → X is weakly compact iff inf Tf − Tg : {f , g } ∈ [C ]2 = 0 for every uncountable -chain C . K. P. Hart A chain condition for operators from C (K )-spaces
  • 29. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesWhy ‘uncountable’ ? Well, . . . Theorem If K is extremally disconnected then T : C (K ) → X is weakly compact iff inf Tf − Tg : {f , g } ∈ [C ]2 = 0 for every uncountable -chain C . In fact if T is not weakly compact then we can find a -chain isomorphic to R where the infimum is positive, that is, there are a δ > 0 and a δ -chain isomorphic to R. K. P. Hart A chain condition for operators from C (K )-spaces
  • 30. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains Sources-chains are easyUncountable -chains are quite ubiquitous: K. P. Hart A chain condition for operators from C (K )-spaces
  • 31. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains Sources-chains are easyUncountable -chains are quite ubiquitous:ExampleThere is an uncountable -chain in C [0, 1] . K. P. Hart A chain condition for operators from C (K )-spaces
  • 32. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains Sources-chains are easyUncountable -chains are quite ubiquitous:ExampleThere is an uncountable -chain in C [0, 1] .Start with f : x → d(x, C), where C is the Cantor set. K. P. Hart A chain condition for operators from C (K )-spaces
  • 33. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains Sources-chains are easyUncountable -chains are quite ubiquitous:ExampleThere is an uncountable -chain in C [0, 1] .Start with f : x → d(x, C), where C is the Cantor set.For t ∈ C let ft = f · χ[0,t] K. P. Hart A chain condition for operators from C (K )-spaces
  • 34. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains Sources-chains are easyUncountable -chains are quite ubiquitous:ExampleThere is an uncountable -chain in C [0, 1] .Start with f : x → d(x, C), where C is the Cantor set.For t ∈ C let ft = f · χ[0,t] , then {ft : t ∈ C} is a -chain. K. P. Hart A chain condition for operators from C (K )-spaces
  • 35. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains Sources-chains are easyUncountable -chains are quite ubiquitous:ExampleThere is an uncountable -chain in C [0, 1] .Start with f : x → d(x, C), where C is the Cantor set.For t ∈ C let ft = f · χ[0,t] , then {ft : t ∈ C} is a -chain.Do we need an instructive picture? K. P. Hart A chain condition for operators from C (K )-spaces
  • 36. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains Sources-chains are easyUncountable -chains are quite ubiquitous:ExampleThere is an uncountable -chain in C [0, 1] .Start with f : x → d(x, C), where C is the Cantor set.For t ∈ C let ft = f · χ[0,t] , then {ft : t ∈ C} is a -chain.Do we need an instructive picture? f2 3 K. P. Hart A chain condition for operators from C (K )-spaces
  • 37. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesOutline 1 Weakly compact operators 2 A chain condition 3 Spaces with and without uncountable δ -chains 4 Sources K. P. Hart A chain condition for operators from C (K )-spaces
  • 38. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesB is not an antichain condition The separable(!) double-arrow space A has a 1 -chain that is isomorphic to R. K. P. Hart A chain condition for operators from C (K )-spaces
  • 39. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesB is not an antichain condition The separable(!) double-arrow space A has a 1 -chain that is isomorphic to R. Remember: we have A = (0, 1] × {0} ∪ [0, 1) × {1} ordered lexicographically. K. P. Hart A chain condition for operators from C (K )-spaces
  • 40. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesB is not an antichain condition The separable(!) double-arrow space A has a 1 -chain that is isomorphic to R. Remember: we have A = (0, 1] × {0} ∪ [0, 1) × {1} ordered lexicographically. For t ∈ (0, 1) let ft be the characteristic function of the interval 0, 1 , t, 0 . K. P. Hart A chain condition for operators from C (K )-spaces
  • 41. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesB is not an antichain condition The separable(!) double-arrow space A has a 1 -chain that is isomorphic to R. Remember: we have A = (0, 1] × {0} ∪ [0, 1) × {1} ordered lexicographically. For t ∈ (0, 1) let ft be the characteristic function of the interval 0, 1 , t, 0 . Time for another instructive picture. K. P. Hart A chain condition for operators from C (K )-spaces
  • 42. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesA few observations Let C be a -chain; for f ∈ C put S(f , C ) = {x : f (x) = 0} {supp g : g ∈ C , g f} K. P. Hart A chain condition for operators from C (K )-spaces
  • 43. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesA few observations Let C be a -chain; for f ∈ C put S(f , C ) = {x : f (x) = 0} {supp g : g ∈ C , g f} Note: in the example in C [0, 1] there are ft , e.g. f 1 , with 3 S(ft ) = ∅, whereas S(f 2 ) = ( 1 , 2 ). 3 33 K. P. Hart A chain condition for operators from C (K )-spaces
  • 44. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesA few observations Let C be a -chain; for f ∈ C put S(f , C ) = {x : f (x) = 0} {supp g : g ∈ C , g f} Note: in the example in C [0, 1] there are ft , e.g. f 1 , with 3 S(ft ) = ∅, whereas S(f 2 ) = ( 1 , 2 ). 3 33 In the chain in C (A) we have S(ft ) = t, 0 for all t. K. P. Hart A chain condition for operators from C (K )-spaces
  • 45. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesA useful lemma From now on all functions are positive. K. P. Hart A chain condition for operators from C (K )-spaces
  • 46. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesA useful lemma From now on all functions are positive. Lemma If C is a δ -chain for some δ > 0 then S(f , C ) = ∅ for all f ∈ C K. P. Hart A chain condition for operators from C (K )-spaces
  • 47. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesA useful lemma From now on all functions are positive. Lemma If C is a δ -chain for some δ > 0 then S(f , C ) = ∅ for all f ∈ C ; in fact there is x ∈ S(f , C ) with f (x) δ. K. P. Hart A chain condition for operators from C (K )-spaces
  • 48. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesA useful lemma From now on all functions are positive. Lemma If C is a δ -chain for some δ > 0 then S(f , C ) = ∅ for all f ∈ C ; in fact there is x ∈ S(f , C ) with f (x) δ. Proof. Clear if f has a direct predecessor. K. P. Hart A chain condition for operators from C (K )-spaces
  • 49. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesA useful lemma From now on all functions are positive. Lemma If C is a δ -chain for some δ > 0 then S(f , C ) = ∅ for all f ∈ C ; in fact there is x ∈ S(f , C ) with f (x) δ. Proof. Clear if f has a direct predecessor. Otherwise let gα : α < θ be increasing and cofinal in {g ∈ C : g f }. K. P. Hart A chain condition for operators from C (K )-spaces
  • 50. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesA useful lemma From now on all functions are positive. Lemma If C is a δ -chain for some δ > 0 then S(f , C ) = ∅ for all f ∈ C ; in fact there is x ∈ S(f , C ) with f (x) δ. Proof. Clear if f has a direct predecessor. Otherwise let gα : α < θ be increasing and cofinal in {g ∈ C : g f }. Pick xα ∈ supp gα+1 supp gα with gα+1 (x) δ. K. P. Hart A chain condition for operators from C (K )-spaces
  • 51. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesA useful lemma From now on all functions are positive. Lemma If C is a δ -chain for some δ > 0 then S(f , C ) = ∅ for all f ∈ C ; in fact there is x ∈ S(f , C ) with f (x) δ. Proof. Clear if f has a direct predecessor. Otherwise let gα : α < θ be increasing and cofinal in {g ∈ C : g f }. Pick xα ∈ supp gα+1 supp gα with gα+1 (x) δ. Any cluster point, x, of gα : α < θ will satisfy f (x) δ K. P. Hart A chain condition for operators from C (K )-spaces
  • 52. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesA useful lemma From now on all functions are positive. Lemma If C is a δ -chain for some δ > 0 then S(f , C ) = ∅ for all f ∈ C ; in fact there is x ∈ S(f , C ) with f (x) δ. Proof. Clear if f has a direct predecessor. Otherwise let gα : α < θ be increasing and cofinal in {g ∈ C : g f }. Pick xα ∈ supp gα+1 supp gα with gα+1 (x) δ. Any cluster point, x, of gα : α < θ will satisfy f (x) δ and g (x) = 0 for all g f . K. P. Hart A chain condition for operators from C (K )-spaces
  • 53. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesThe convergent sequence C (ω + 1) has an uncountable -chain. K. P. Hart A chain condition for operators from C (K )-spaces
  • 54. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesThe convergent sequence C (ω + 1) has an uncountable -chain. Let b : ω → Q be a bijection. For t ∈ R define ft by 2−α if b(α) < t ft (α) = 0 otherwise. K. P. Hart A chain condition for operators from C (K )-spaces
  • 55. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesThe convergent sequence C (ω + 1) has an uncountable -chain. Let b : ω → Q be a bijection. For t ∈ R define ft by 2−α if b(α) < t ft (α) = 0 otherwise. If δ > 0 then every δ -chain in C (ω + 1) is countable. K. P. Hart A chain condition for operators from C (K )-spaces
  • 56. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesAnother lemma Lemma If K is locally connected and if C is a δ -chain for some δ > 0 then S(f , C ) is (nonempty and) open. K. P. Hart A chain condition for operators from C (K )-spaces
  • 57. Weakly compact operators A chain condition Spaces with and without uncountable δ -chains SourcesAnother lemma Lemma If K is locally connected and if C is a δ -chain for some δ > 0 then S(f , C ) is (nonempty and) open. Proof. Let x ∈ S(f , C ) and let U be a connected neighbourhood of x 1 such that f (y ) > 2 f (x) for all y ∈ U. We claim U ∩ supp g = ∅ if g f. K. P. Hart A chain condition for operators from C (K )-spaces
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