Ideals in L ( X ) A maximal algebraic ideal is automatically closed since the invertible elements in a Banach algebra form an open set, so every (always proper) closed ideal is contained in a closed maximal ideal. What are the maximal ones? Is there even a largest ideal? Let M ( X ) denote all operators T on X s.t. the identity operator I X does not factor through T ( I X � = BTA ). It is obvious that M ( X ) is an ideal in L ( X ) if it is closed under addition, in which case it clearly is the largest ideal in L ( X ) . It is known, but non trivial, that M ( L p ) is closed under addition, and also that M ( L p ) is the set of L p -singular operators, that is the set of operators that are not an isomorphism when restricted to any subspace isomorphic to L p . [Enflo, Starbird ’79] for p = 1; [Johnson, Maurey, S, Tzafriri ’79] for 1 < p � = 2 < ∞ . Gideon Schechtman Ideals in L ( L p )
Ideals in L ( X ) A maximal algebraic ideal is automatically closed since the invertible elements in a Banach algebra form an open set, so every (always proper) closed ideal is contained in a closed maximal ideal. What are the maximal ones? Is there even a largest ideal? Let M ( X ) denote all operators T on X s.t. the identity operator I X does not factor through T ( I X � = BTA ). It is obvious that M ( X ) is an ideal in L ( X ) if it is closed under addition, in which case it clearly is the largest ideal in L ( X ) . It is known, but non trivial, that M ( L p ) is closed under addition, and also that M ( L p ) is the set of L p -singular operators, that is the set of operators that are not an isomorphism when restricted to any subspace isomorphic to L p . [Enflo, Starbird ’79] for p = 1; [Johnson, Maurey, S, Tzafriri ’79] for 1 < p � = 2 < ∞ . Gideon Schechtman Ideals in L ( L p )
ideals in L ( X ) A common way of constructing a (not necessarily closed) ideal in L ( X ) is to take some operator U : Y → Z between Banach spaces and let I U be the collection of all operators on X that factor through U , i.e., all T ∈ L ( X ) s.t. there exist A ∈ L ( X , Y ) and B ∈ L ( Z , X ) s.t. T = BUA . L ( X ) I U L ( X ) ⊂ I U is clear, so I U is an ideal in L ( X ) if I U is closed under addition. One usually guarantees this by using a U s.t. U ⊕ U : Y ⊕ Y → Z ⊕ Z factors through U , and these are the only U that I will use. Then the closure I U will be a proper ideal in L ( X ) as long as I X does not factor through U . Gideon Schechtman Ideals in L ( L p )
ideals in L ( X ) A common way of constructing a (not necessarily closed) ideal in L ( X ) is to take some operator U : Y → Z between Banach spaces and let I U be the collection of all operators on X that factor through U , i.e., all T ∈ L ( X ) s.t. there exist A ∈ L ( X , Y ) and B ∈ L ( Z , X ) s.t. T = BUA . L ( X ) I U L ( X ) ⊂ I U is clear, so I U is an ideal in L ( X ) if I U is closed under addition. One usually guarantees this by using a U s.t. U ⊕ U : Y ⊕ Y → Z ⊕ Z factors through U , and these are the only U that I will use. Then the closure I U will be a proper ideal in L ( X ) as long as I X does not factor through U . Gideon Schechtman Ideals in L ( L p )
ideals in L ( X ) A common way of constructing a (not necessarily closed) ideal in L ( X ) is to take some operator U : Y → Z between Banach spaces and let I U be the collection of all operators on X that factor through U , i.e., all T ∈ L ( X ) s.t. there exist A ∈ L ( X , Y ) and B ∈ L ( Z , X ) s.t. T = BUA . L ( X ) I U L ( X ) ⊂ I U is clear, so I U is an ideal in L ( X ) if I U is closed under addition. One usually guarantees this by using a U s.t. U ⊕ U : Y ⊕ Y → Z ⊕ Z factors through U , and these are the only U that I will use. Then the closure I U will be a proper ideal in L ( X ) as long as I X does not factor through U . Gideon Schechtman Ideals in L ( L p )
Large and Small Ideals I U : All T ∈ L ( X ) that factor through U . S ( X ) : Strictly singular operators on X . An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. So, for example, I U is small if U is strictly singular and U ⊕ U factors through U . And, for example, I U is large if U = I Y for some complemented subspace Y of X and Y ⊕ Y is isomorphic to Y . To simplify notation, I’ll write I Y instead of I I Y . Gideon Schechtman Ideals in L ( L p )
Large and Small Ideals I U : All T ∈ L ( X ) that factor through U . S ( X ) : Strictly singular operators on X . An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. So, for example, I U is small if U is strictly singular and U ⊕ U factors through U . And, for example, I U is large if U = I Y for some complemented subspace Y of X and Y ⊕ Y is isomorphic to Y . To simplify notation, I’ll write I Y instead of I I Y . Gideon Schechtman Ideals in L ( L p )
Large and Small Ideals I U : All T ∈ L ( X ) that factor through U . S ( X ) : Strictly singular operators on X . An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. So, for example, I U is small if U is strictly singular and U ⊕ U factors through U . And, for example, I U is large if U = I Y for some complemented subspace Y of X and Y ⊕ Y is isomorphic to Y . To simplify notation, I’ll write I Y instead of I I Y . Gideon Schechtman Ideals in L ( L p )
Large and Small Ideals I U : All T ∈ L ( X ) that factor through U . S ( X ) : Strictly singular operators on X . An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. So, for example, I U is small if U is strictly singular and U ⊕ U factors through U . And, for example, I U is large if U = I Y for some complemented subspace Y of X and Y ⊕ Y is isomorphic to Y . To simplify notation, I’ll write I Y instead of I I Y . Gideon Schechtman Ideals in L ( L p )
Large and Small Ideals I U : All T ∈ L ( X ) that factor through U . S ( X ) : Strictly singular operators on X . An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. So, for example, I U is small if U is strictly singular and U ⊕ U factors through U . And, for example, I U is large if U = I Y for some complemented subspace Y of X and Y ⊕ Y is isomorphic to Y . To simplify notation, I’ll write I Y instead of I I Y . Gideon Schechtman Ideals in L ( L p )
Large and Small Ideals I U : All T ∈ L ( X ) that factor through U . S ( X ) : Strictly singular operators on X . An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. So, for example, I U is small if U is strictly singular and U ⊕ U factors through U . And, for example, I U is large if U = I Y for some complemented subspace Y of X and Y ⊕ Y is isomorphic to Y . To simplify notation, I’ll write I Y instead of I I Y . Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. Small closed ideals in L ( L 1 ) include K ( L 1 ) , S ( L 1 ) , and W ( L 1 ) . But W ( L 1 ) = S ( L 1 ) Dunford-Pettis property of L 1 . Large closed ideals in L ( L 1 ) include I ℓ 1 and the largest ideal M ( L 1 ) (and also the Dunford–Pettis opertors). Incidently, Every large ideal in L ( L 1 ) contains I ℓ 1 and I ℓ 1 contains any small ideal in L ( L 1 ) . Until recently this is all that were known. This led Pietsch to ask in his 1979 book “Operator Ideals" whether there are infinitely many closed ideals in L ( L 1 ) . Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. Small closed ideals in L ( L 1 ) include K ( L 1 ) , S ( L 1 ) , and W ( L 1 ) . But W ( L 1 ) = S ( L 1 ) Dunford-Pettis property of L 1 . Large closed ideals in L ( L 1 ) include I ℓ 1 and the largest ideal M ( L 1 ) (and also the Dunford–Pettis opertors). Incidently, Every large ideal in L ( L 1 ) contains I ℓ 1 and I ℓ 1 contains any small ideal in L ( L 1 ) . Until recently this is all that were known. This led Pietsch to ask in his 1979 book “Operator Ideals" whether there are infinitely many closed ideals in L ( L 1 ) . Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. Small closed ideals in L ( L 1 ) include K ( L 1 ) , S ( L 1 ) , and W ( L 1 ) . But W ( L 1 ) = S ( L 1 ) Dunford-Pettis property of L 1 . Large closed ideals in L ( L 1 ) include I ℓ 1 and the largest ideal M ( L 1 ) (and also the Dunford–Pettis opertors). Incidently, Every large ideal in L ( L 1 ) contains I ℓ 1 and I ℓ 1 contains any small ideal in L ( L 1 ) . Until recently this is all that were known. This led Pietsch to ask in his 1979 book “Operator Ideals" whether there are infinitely many closed ideals in L ( L 1 ) . Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. Small closed ideals in L ( L 1 ) include K ( L 1 ) , S ( L 1 ) , and W ( L 1 ) . But W ( L 1 ) = S ( L 1 ) Dunford-Pettis property of L 1 . Large closed ideals in L ( L 1 ) include I ℓ 1 and the largest ideal M ( L 1 ) (and also the Dunford–Pettis opertors). Incidently, Every large ideal in L ( L 1 ) contains I ℓ 1 and I ℓ 1 contains any small ideal in L ( L 1 ) . Until recently this is all that were known. This led Pietsch to ask in his 1979 book “Operator Ideals" whether there are infinitely many closed ideals in L ( L 1 ) . Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. Small closed ideals in L ( L 1 ) include K ( L 1 ) , S ( L 1 ) , and W ( L 1 ) . But W ( L 1 ) = S ( L 1 ) Dunford-Pettis property of L 1 . Large closed ideals in L ( L 1 ) include I ℓ 1 and the largest ideal M ( L 1 ) (and also the Dunford–Pettis opertors). Incidently, Every large ideal in L ( L 1 ) contains I ℓ 1 and I ℓ 1 contains any small ideal in L ( L 1 ) . Until recently this is all that were known. This led Pietsch to ask in his 1979 book “Operator Ideals" whether there are infinitely many closed ideals in L ( L 1 ) . Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) - the difficulty It is easy to build closed ideals in L ( X ) ; in particular, in L ( L 1 ) ; but difficult to prove that ideals are different. For example, for 1 < p < ∞ , let I L p be the (non closed) ideal of operators on L 1 that factor through L p . These are all different, but their closures I L p are all equal to the weakly compact operators on L 1 . One would guess that the key to solving Pietsch’s problem was to find just one new closed ideal in L ( L 1 ) . A couple of years ago Bill and I did that. The ideal is the closure of I J 2 , where J 2 : ℓ 1 → L 1 maps the unit vector basis of ℓ 1 onto the Rademacher functions IID Bernoulli random variables that take on the values 1 and − 1, each with probability 1 / 2 . We were excited when we were able to prove that I J 2 is different from the previously known ideals. We then looked at I J p , 1 < p < 2, where J p : ℓ 1 → L 1 maps the unit vector basis of ℓ 1 onto IID p -stable random variables. The ideals I J p are all different, but it turns out that all the I J p are equal to I J 2 ! Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) - the difficulty It is easy to build closed ideals in L ( X ) ; in particular, in L ( L 1 ) ; but difficult to prove that ideals are different. For example, for 1 < p < ∞ , let I L p be the (non closed) ideal of operators on L 1 that factor through L p . These are all different, but their closures I L p are all equal to the weakly compact operators on L 1 . One would guess that the key to solving Pietsch’s problem was to find just one new closed ideal in L ( L 1 ) . A couple of years ago Bill and I did that. The ideal is the closure of I J 2 , where J 2 : ℓ 1 → L 1 maps the unit vector basis of ℓ 1 onto the Rademacher functions IID Bernoulli random variables that take on the values 1 and − 1, each with probability 1 / 2 . We were excited when we were able to prove that I J 2 is different from the previously known ideals. We then looked at I J p , 1 < p < 2, where J p : ℓ 1 → L 1 maps the unit vector basis of ℓ 1 onto IID p -stable random variables. The ideals I J p are all different, but it turns out that all the I J p are equal to I J 2 ! Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) - the difficulty It is easy to build closed ideals in L ( X ) ; in particular, in L ( L 1 ) ; but difficult to prove that ideals are different. For example, for 1 < p < ∞ , let I L p be the (non closed) ideal of operators on L 1 that factor through L p . These are all different, but their closures I L p are all equal to the weakly compact operators on L 1 . One would guess that the key to solving Pietsch’s problem was to find just one new closed ideal in L ( L 1 ) . A couple of years ago Bill and I did that. The ideal is the closure of I J 2 , where J 2 : ℓ 1 → L 1 maps the unit vector basis of ℓ 1 onto the Rademacher functions IID Bernoulli random variables that take on the values 1 and − 1, each with probability 1 / 2 . We were excited when we were able to prove that I J 2 is different from the previously known ideals. We then looked at I J p , 1 < p < 2, where J p : ℓ 1 → L 1 maps the unit vector basis of ℓ 1 onto IID p -stable random variables. The ideals I J p are all different, but it turns out that all the I J p are equal to I J 2 ! Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) - the difficulty It is easy to build closed ideals in L ( X ) ; in particular, in L ( L 1 ) ; but difficult to prove that ideals are different. For example, for 1 < p < ∞ , let I L p be the (non closed) ideal of operators on L 1 that factor through L p . These are all different, but their closures I L p are all equal to the weakly compact operators on L 1 . One would guess that the key to solving Pietsch’s problem was to find just one new closed ideal in L ( L 1 ) . A couple of years ago Bill and I did that. The ideal is the closure of I J 2 , where J 2 : ℓ 1 → L 1 maps the unit vector basis of ℓ 1 onto the Rademacher functions IID Bernoulli random variables that take on the values 1 and − 1, each with probability 1 / 2 . We were excited when we were able to prove that I J 2 is different from the previously known ideals. We then looked at I J p , 1 < p < 2, where J p : ℓ 1 → L 1 maps the unit vector basis of ℓ 1 onto IID p -stable random variables. The ideals I J p are all different, but it turns out that all the I J p are equal to I J 2 ! Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) - the difficulty It is easy to build closed ideals in L ( X ) ; in particular, in L ( L 1 ) ; but difficult to prove that ideals are different. For example, for 1 < p < ∞ , let I L p be the (non closed) ideal of operators on L 1 that factor through L p . These are all different, but their closures I L p are all equal to the weakly compact operators on L 1 . One would guess that the key to solving Pietsch’s problem was to find just one new closed ideal in L ( L 1 ) . A couple of years ago Bill and I did that. The ideal is the closure of I J 2 , where J 2 : ℓ 1 → L 1 maps the unit vector basis of ℓ 1 onto the Rademacher functions IID Bernoulli random variables that take on the values 1 and − 1, each with probability 1 / 2 . We were excited when we were able to prove that I J 2 is different from the previously known ideals. We then looked at I J p , 1 < p < 2, where J p : ℓ 1 → L 1 maps the unit vector basis of ℓ 1 onto IID p -stable random variables. The ideals I J p are all different, but it turns out that all the I J p are equal to I J 2 ! Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) - the difficulty It is easy to build closed ideals in L ( X ) ; in particular, in L ( L 1 ) ; but difficult to prove that ideals are different. For example, for 1 < p < ∞ , let I L p be the (non closed) ideal of operators on L 1 that factor through L p . These are all different, but their closures I L p are all equal to the weakly compact operators on L 1 . One would guess that the key to solving Pietsch’s problem was to find just one new closed ideal in L ( L 1 ) . A couple of years ago Bill and I did that. The ideal is the closure of I J 2 , where J 2 : ℓ 1 → L 1 maps the unit vector basis of ℓ 1 onto the Rademacher functions IID Bernoulli random variables that take on the values 1 and − 1, each with probability 1 / 2 . We were excited when we were able to prove that I J 2 is different from the previously known ideals. We then looked at I J p , 1 < p < 2, where J p : ℓ 1 → L 1 maps the unit vector basis of ℓ 1 onto IID p -stable random variables. The ideals I J p are all different, but it turns out that all the I J p are equal to I J 2 ! Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) Theorem. [JPS] There are at least 2 ℵ 0 (small) closed ideals in L ( L 1 ) . It remains open whether there are infinitely many large closed ideals in L ( L 1 ) . This is connected to the unsolved problem whether every infinite dimensional complemented subspace of L 1 is isomorphic either to ℓ 1 or to L 1 . Also open is whether there are more than 2 ℵ 0 closed ideals in L ( L 1 ) . The new ideals are a familty ( I U q ) 2 < q < ∞ , where U q : ℓ 1 → L 1 {− 1 , 1 } N maps the unit vector basis of ℓ 1 to a carefully chosen Λ( q ) -set of characters. (A set of characters is Λ( q ) if the L 1 norm is equivalent to the L q norm on their linear span.) Bourgain’s solution to Rudin’s Λ( q ) -set problem is used (could be avoided by using B-space theory results from the 1970s) . The problem is to show that these ideals are all different. Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) Theorem. [JPS] There are at least 2 ℵ 0 (small) closed ideals in L ( L 1 ) . It remains open whether there are infinitely many large closed ideals in L ( L 1 ) . This is connected to the unsolved problem whether every infinite dimensional complemented subspace of L 1 is isomorphic either to ℓ 1 or to L 1 . Also open is whether there are more than 2 ℵ 0 closed ideals in L ( L 1 ) . The new ideals are a familty ( I U q ) 2 < q < ∞ , where U q : ℓ 1 → L 1 {− 1 , 1 } N maps the unit vector basis of ℓ 1 to a carefully chosen Λ( q ) -set of characters. (A set of characters is Λ( q ) if the L 1 norm is equivalent to the L q norm on their linear span.) Bourgain’s solution to Rudin’s Λ( q ) -set problem is used (could be avoided by using B-space theory results from the 1970s) . The problem is to show that these ideals are all different. Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) Theorem. [JPS] There are at least 2 ℵ 0 (small) closed ideals in L ( L 1 ) . It remains open whether there are infinitely many large closed ideals in L ( L 1 ) . This is connected to the unsolved problem whether every infinite dimensional complemented subspace of L 1 is isomorphic either to ℓ 1 or to L 1 . Also open is whether there are more than 2 ℵ 0 closed ideals in L ( L 1 ) . The new ideals are a familty ( I U q ) 2 < q < ∞ , where U q : ℓ 1 → L 1 {− 1 , 1 } N maps the unit vector basis of ℓ 1 to a carefully chosen Λ( q ) -set of characters. (A set of characters is Λ( q ) if the L 1 norm is equivalent to the L q norm on their linear span.) Bourgain’s solution to Rudin’s Λ( q ) -set problem is used (could be avoided by using B-space theory results from the 1970s) . The problem is to show that these ideals are all different. Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) Theorem. [JPS] There are at least 2 ℵ 0 (small) closed ideals in L ( L 1 ) . It remains open whether there are infinitely many large closed ideals in L ( L 1 ) . This is connected to the unsolved problem whether every infinite dimensional complemented subspace of L 1 is isomorphic either to ℓ 1 or to L 1 . Also open is whether there are more than 2 ℵ 0 closed ideals in L ( L 1 ) . The new ideals are a familty ( I U q ) 2 < q < ∞ , where U q : ℓ 1 → L 1 {− 1 , 1 } N maps the unit vector basis of ℓ 1 to a carefully chosen Λ( q ) -set of characters. (A set of characters is Λ( q ) if the L 1 norm is equivalent to the L q norm on their linear span.) Bourgain’s solution to Rudin’s Λ( q ) -set problem is used (could be avoided by using B-space theory results from the 1970s) . The problem is to show that these ideals are all different. Gideon Schechtman Ideals in L ( L p )
Ideals in L ( L 1 ) Theorem. [JPS] There are at least 2 ℵ 0 (small) closed ideals in L ( L 1 ) . It remains open whether there are infinitely many large closed ideals in L ( L 1 ) . This is connected to the unsolved problem whether every infinite dimensional complemented subspace of L 1 is isomorphic either to ℓ 1 or to L 1 . Also open is whether there are more than 2 ℵ 0 closed ideals in L ( L 1 ) . The new ideals are a familty ( I U q ) 2 < q < ∞ , where U q : ℓ 1 → L 1 {− 1 , 1 } N maps the unit vector basis of ℓ 1 to a carefully chosen Λ( q ) -set of characters. (A set of characters is Λ( q ) if the L 1 norm is equivalent to the L q norm on their linear span.) Bourgain’s solution to Rudin’s Λ( q ) -set problem is used (could be avoided by using B-space theory results from the 1970s) . The problem is to show that these ideals are all different. Gideon Schechtman Ideals in L ( L p )
Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. [S ’75] There are infinitely many isomorphically different complemented subspaces of L p , each isomorphic to its square, hence there are infinitely many (large) closed ideals in L ( L p ) . [Bourgain, Rosenthal, S ’81] There are ℵ 1 isomorphically different complemented subspaces of L p , each isomorphic to its square, hence there are ℵ 1 (large) closed ideals in L ( L p ) . This leaves open whether there are there more than ℵ 1 (large?/small?) closed ideals in L ( L p ) ? Maybe there are even 2 2 ℵ 0 (large?/small?) closed ideals. Gideon Schechtman Ideals in L ( L p )
Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. [S ’75] There are infinitely many isomorphically different complemented subspaces of L p , each isomorphic to its square, hence there are infinitely many (large) closed ideals in L ( L p ) . [Bourgain, Rosenthal, S ’81] There are ℵ 1 isomorphically different complemented subspaces of L p , each isomorphic to its square, hence there are ℵ 1 (large) closed ideals in L ( L p ) . This leaves open whether there are there more than ℵ 1 (large?/small?) closed ideals in L ( L p ) ? Maybe there are even 2 2 ℵ 0 (large?/small?) closed ideals. Gideon Schechtman Ideals in L ( L p )
Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. [S ’75] There are infinitely many isomorphically different complemented subspaces of L p , each isomorphic to its square, hence there are infinitely many (large) closed ideals in L ( L p ) . [Bourgain, Rosenthal, S ’81] There are ℵ 1 isomorphically different complemented subspaces of L p , each isomorphic to its square, hence there are ℵ 1 (large) closed ideals in L ( L p ) . This leaves open whether there are there more than ℵ 1 (large?/small?) closed ideals in L ( L p ) ? Maybe there are even 2 2 ℵ 0 (large?/small?) closed ideals. Gideon Schechtman Ideals in L ( L p )
Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ An ideal I is small if I ⊂ S ( X ) ; otherwise it is large. [S ’75] There are infinitely many isomorphically different complemented subspaces of L p , each isomorphic to its square, hence there are infinitely many (large) closed ideals in L ( L p ) . [Bourgain, Rosenthal, S ’81] There are ℵ 1 isomorphically different complemented subspaces of L p , each isomorphic to its square, hence there are ℵ 1 (large) closed ideals in L ( L p ) . This leaves open whether there are there more than ℵ 1 (large?/small?) closed ideals in L ( L p ) ? Maybe there are even 2 2 ℵ 0 (large?/small?) closed ideals. Gideon Schechtman Ideals in L ( L p )
Small Ideals in L ( L p ) , 1 < p � = 2 < ∞ The following solved the first problem for small ideals Theorem. (Schlumprecht,Zsak ’18) There are infinitely many; in fact, at least 2 ℵ 0 ; (small) closed ideals in L ( L p ) , 1 < p � = 2 < ∞ . The ideals constructed in [SZ ’18] are all of the form I U with U a basis to basis mapping from ℓ r to ℓ s but the bases for ℓ r , ℓ s are not the standard unit vector basis. Whether there are more than 2 ℵ 0 small closed ideals in L ( L p ) remains open. But, Gideon Schechtman Ideals in L ( L p )
Small Ideals in L ( L p ) , 1 < p � = 2 < ∞ The following solved the first problem for small ideals Theorem. (Schlumprecht,Zsak ’18) There are infinitely many; in fact, at least 2 ℵ 0 ; (small) closed ideals in L ( L p ) , 1 < p � = 2 < ∞ . The ideals constructed in [SZ ’18] are all of the form I U with U a basis to basis mapping from ℓ r to ℓ s but the bases for ℓ r , ℓ s are not the standard unit vector basis. Whether there are more than 2 ℵ 0 small closed ideals in L ( L p ) remains open. But, Gideon Schechtman Ideals in L ( L p )
Small Ideals in L ( L p ) , 1 < p � = 2 < ∞ The following solved the first problem for small ideals Theorem. (Schlumprecht,Zsak ’18) There are infinitely many; in fact, at least 2 ℵ 0 ; (small) closed ideals in L ( L p ) , 1 < p � = 2 < ∞ . The ideals constructed in [SZ ’18] are all of the form I U with U a basis to basis mapping from ℓ r to ℓ s but the bases for ℓ r , ℓ s are not the standard unit vector basis. Whether there are more than 2 ℵ 0 small closed ideals in L ( L p ) remains open. But, Gideon Schechtman Ideals in L ( L p )
Small Ideals in L ( L p ) , 1 < p � = 2 < ∞ The following solved the first problem for small ideals Theorem. (Schlumprecht,Zsak ’18) There are infinitely many; in fact, at least 2 ℵ 0 ; (small) closed ideals in L ( L p ) , 1 < p � = 2 < ∞ . The ideals constructed in [SZ ’18] are all of the form I U with U a basis to basis mapping from ℓ r to ℓ s but the bases for ℓ r , ℓ s are not the standard unit vector basis. Whether there are more than 2 ℵ 0 small closed ideals in L ( L p ) remains open. But, Gideon Schechtman Ideals in L ( L p )
Small Ideals in L ( L p ) , 1 < p � = 2 < ∞ The following solved the first problem for small ideals Theorem. (Schlumprecht,Zsak ’18) There are infinitely many; in fact, at least 2 ℵ 0 ; (small) closed ideals in L ( L p ) , 1 < p � = 2 < ∞ . The ideals constructed in [SZ ’18] are all of the form I U with U a basis to basis mapping from ℓ r to ℓ s but the bases for ℓ r , ℓ s are not the standard unit vector basis. Whether there are more than 2 ℵ 0 small closed ideals in L ( L p ) remains open. But, Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ We recently proved, Theorem. (JS ’19) There are 2 2 ℵ 0 ; (large) closed ideals in L ( L p ) , 1 < p � = 2 < ∞ . The proof relays on fine properties of spaces spanned by independent random variables in L p , 2 < p < ∞ , a topic investigated mostly by Rosenthal in the 1970-s. Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ We recently proved, Theorem. (JS ’19) There are 2 2 ℵ 0 ; (large) closed ideals in L ( L p ) , 1 < p � = 2 < ∞ . The proof relays on fine properties of spaces spanned by independent random variables in L p , 2 < p < ∞ , a topic investigated mostly by Rosenthal in the 1970-s. Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ Recall that for a sequence u = { u j } ∞ j = 1 of positive real numbers and for p > 2, the Banach space X p , u is the real sequence space with norm ∞ ∞ � � �{ a j } ∞ | a j | p ) 1 / p , ( | a j u j | 2 ) 1 / 2 } . j = 1 � = max { ( j = 1 j = 1 Rosenthal proved that X p , u is isomorphic to a complemented subspace of L p with the isomorphism constant and the complementation constant depending only on p . 2 p If u is such that lim j → 0 u j = 0 but � ∞ p − 2 = ∞ then one j = 1 | u j | gets a space isomorphically different from ℓ p , ℓ 2 and ℓ p ⊕ ℓ 2 . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ Recall that for a sequence u = { u j } ∞ j = 1 of positive real numbers and for p > 2, the Banach space X p , u is the real sequence space with norm ∞ ∞ � � �{ a j } ∞ | a j | p ) 1 / p , ( | a j u j | 2 ) 1 / 2 } . j = 1 � = max { ( j = 1 j = 1 Rosenthal proved that X p , u is isomorphic to a complemented subspace of L p with the isomorphism constant and the complementation constant depending only on p . 2 p If u is such that lim j → 0 u j = 0 but � ∞ p − 2 = ∞ then one j = 1 | u j | gets a space isomorphically different from ℓ p , ℓ 2 and ℓ p ⊕ ℓ 2 . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ Recall that for a sequence u = { u j } ∞ j = 1 of positive real numbers and for p > 2, the Banach space X p , u is the real sequence space with norm ∞ ∞ � � �{ a j } ∞ | a j | p ) 1 / p , ( | a j u j | 2 ) 1 / 2 } . j = 1 � = max { ( j = 1 j = 1 Rosenthal proved that X p , u is isomorphic to a complemented subspace of L p with the isomorphism constant and the complementation constant depending only on p . 2 p If u is such that lim j → 0 u j = 0 but � ∞ p − 2 = ∞ then one j = 1 | u j | gets a space isomorphically different from ℓ p , ℓ 2 and ℓ p ⊕ ℓ 2 . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ Recall that for a sequence u = { u j } ∞ j = 1 of positive real numbers and for p > 2, the Banach space X p , u is the real sequence space with norm ∞ ∞ � � �{ a j } ∞ | a j | p ) 1 / p , ( | a j u j | 2 ) 1 / 2 } . j = 1 � = max { ( j = 1 j = 1 Rosenthal proved that X p , u is isomorphic to a complemented subspace of L p with the isomorphism constant and the complementation constant depending only on p . 2 p If u is such that lim j → 0 u j = 0 but � ∞ p − 2 = ∞ then one j = 1 | u j | gets a space isomorphically different from ℓ p , ℓ 2 and ℓ p ⊕ ℓ 2 . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ j = 1 � X p , u = max { ( � ∞ j = 1 | a j | p ) 1 / p , ( � ∞ �{ a j } ∞ j = 1 | a j u j | 2 ) 1 / 2 } . However, for different u satisfying the two conditions above the different X p , u spaces are mutually isomorphic. We denote by X p any of these spaces. We’ll need more properties of the spaces X p , u but right now we only need the representation above and we think of X p , u as a subspace of ℓ p ⊕ ∞ ℓ 2 . Let { e j } ∞ j = 1 be the unit vector basis of ℓ p and { f j } ∞ j = 1 be the unit vector basis of ℓ 2 . Let v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 be two positive real sequences such that δ j = w j / v j → 0 as j → ∞ . Set g v g w j = e j + v j f j ∈ ℓ p ⊕ ∞ ℓ 2 and = e j + w j f j ∈ ℓ p ⊕ ∞ ℓ 2 . j Then { g v j } ∞ j = 1 is the unit vector basis of X p , v and { g w j } ∞ j = 1 is the unit vector basis of X p , w . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ j = 1 � X p , u = max { ( � ∞ j = 1 | a j | p ) 1 / p , ( � ∞ �{ a j } ∞ j = 1 | a j u j | 2 ) 1 / 2 } . However, for different u satisfying the two conditions above the different X p , u spaces are mutually isomorphic. We denote by X p any of these spaces. We’ll need more properties of the spaces X p , u but right now we only need the representation above and we think of X p , u as a subspace of ℓ p ⊕ ∞ ℓ 2 . Let { e j } ∞ j = 1 be the unit vector basis of ℓ p and { f j } ∞ j = 1 be the unit vector basis of ℓ 2 . Let v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 be two positive real sequences such that δ j = w j / v j → 0 as j → ∞ . Set g v g w j = e j + v j f j ∈ ℓ p ⊕ ∞ ℓ 2 and = e j + w j f j ∈ ℓ p ⊕ ∞ ℓ 2 . j Then { g v j } ∞ j = 1 is the unit vector basis of X p , v and { g w j } ∞ j = 1 is the unit vector basis of X p , w . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ j = 1 � X p , u = max { ( � ∞ j = 1 | a j | p ) 1 / p , ( � ∞ �{ a j } ∞ j = 1 | a j u j | 2 ) 1 / 2 } . However, for different u satisfying the two conditions above the different X p , u spaces are mutually isomorphic. We denote by X p any of these spaces. We’ll need more properties of the spaces X p , u but right now we only need the representation above and we think of X p , u as a subspace of ℓ p ⊕ ∞ ℓ 2 . Let { e j } ∞ j = 1 be the unit vector basis of ℓ p and { f j } ∞ j = 1 be the unit vector basis of ℓ 2 . Let v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 be two positive real sequences such that δ j = w j / v j → 0 as j → ∞ . Set g v g w j = e j + v j f j ∈ ℓ p ⊕ ∞ ℓ 2 and = e j + w j f j ∈ ℓ p ⊕ ∞ ℓ 2 . j Then { g v j } ∞ j = 1 is the unit vector basis of X p , v and { g w j } ∞ j = 1 is the unit vector basis of X p , w . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ j = 1 � X p , u = max { ( � ∞ j = 1 | a j | p ) 1 / p , ( � ∞ �{ a j } ∞ j = 1 | a j u j | 2 ) 1 / 2 } . However, for different u satisfying the two conditions above the different X p , u spaces are mutually isomorphic. We denote by X p any of these spaces. We’ll need more properties of the spaces X p , u but right now we only need the representation above and we think of X p , u as a subspace of ℓ p ⊕ ∞ ℓ 2 . Let { e j } ∞ j = 1 be the unit vector basis of ℓ p and { f j } ∞ j = 1 be the unit vector basis of ℓ 2 . Let v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 be two positive real sequences such that δ j = w j / v j → 0 as j → ∞ . Set g v g w j = e j + v j f j ∈ ℓ p ⊕ ∞ ℓ 2 and = e j + w j f j ∈ ℓ p ⊕ ∞ ℓ 2 . j Then { g v j } ∞ j = 1 is the unit vector basis of X p , v and { g w j } ∞ j = 1 is the unit vector basis of X p , w . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ j = 1 � X p , u = max { ( � ∞ j = 1 | a j | p ) 1 / p , ( � ∞ �{ a j } ∞ j = 1 | a j u j | 2 ) 1 / 2 } . However, for different u satisfying the two conditions above the different X p , u spaces are mutually isomorphic. We denote by X p any of these spaces. We’ll need more properties of the spaces X p , u but right now we only need the representation above and we think of X p , u as a subspace of ℓ p ⊕ ∞ ℓ 2 . Let { e j } ∞ j = 1 be the unit vector basis of ℓ p and { f j } ∞ j = 1 be the unit vector basis of ℓ 2 . Let v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 be two positive real sequences such that δ j = w j / v j → 0 as j → ∞ . Set g v g w j = e j + v j f j ∈ ℓ p ⊕ ∞ ℓ 2 and = e j + w j f j ∈ ℓ p ⊕ ∞ ℓ 2 . j Then { g v j } ∞ j = 1 is the unit vector basis of X p , v and { g w j } ∞ j = 1 is the unit vector basis of X p , w . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ j = 1 � X p , u = max { ( � ∞ j = 1 | a j | p ) 1 / p , ( � ∞ �{ a j } ∞ j = 1 | a j u j | 2 ) 1 / 2 } . However, for different u satisfying the two conditions above the different X p , u spaces are mutually isomorphic. We denote by X p any of these spaces. We’ll need more properties of the spaces X p , u but right now we only need the representation above and we think of X p , u as a subspace of ℓ p ⊕ ∞ ℓ 2 . Let { e j } ∞ j = 1 be the unit vector basis of ℓ p and { f j } ∞ j = 1 be the unit vector basis of ℓ 2 . Let v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 be two positive real sequences such that δ j = w j / v j → 0 as j → ∞ . Set g v g w j = e j + v j f j ∈ ℓ p ⊕ ∞ ℓ 2 and = e j + w j f j ∈ ℓ p ⊕ ∞ ℓ 2 . j Then { g v j } ∞ j = 1 is the unit vector basis of X p , v and { g w j } ∞ j = 1 is the unit vector basis of X p , w . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ j = 1 � X p , u = max { ( � ∞ j = 1 | a j | p ) 1 / p , ( � ∞ �{ a j } ∞ j = 1 | a j u j | 2 ) 1 / 2 } . However, for different u satisfying the two conditions above the different X p , u spaces are mutually isomorphic. We denote by X p any of these spaces. We’ll need more properties of the spaces X p , u but right now we only need the representation above and we think of X p , u as a subspace of ℓ p ⊕ ∞ ℓ 2 . Let { e j } ∞ j = 1 be the unit vector basis of ℓ p and { f j } ∞ j = 1 be the unit vector basis of ℓ 2 . Let v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 be two positive real sequences such that δ j = w j / v j → 0 as j → ∞ . Set g v g w j = e j + v j f j ∈ ℓ p ⊕ ∞ ℓ 2 and = e j + w j f j ∈ ℓ p ⊕ ∞ ℓ 2 . j Then { g v j } ∞ j = 1 is the unit vector basis of X p , v and { g w j } ∞ j = 1 is the unit vector basis of X p , w . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ j = 1 � X p , u = max { ( � ∞ j = 1 | a j | p ) 1 / p , ( � ∞ �{ a j } ∞ j = 1 | a j u j | 2 ) 1 / 2 } . However, for different u satisfying the two conditions above the different X p , u spaces are mutually isomorphic. We denote by X p any of these spaces. We’ll need more properties of the spaces X p , u but right now we only need the representation above and we think of X p , u as a subspace of ℓ p ⊕ ∞ ℓ 2 . Let { e j } ∞ j = 1 be the unit vector basis of ℓ p and { f j } ∞ j = 1 be the unit vector basis of ℓ 2 . Let v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 be two positive real sequences such that δ j = w j / v j → 0 as j → ∞ . Set g v g w j = e j + v j f j ∈ ℓ p ⊕ ∞ ℓ 2 and = e j + w j f j ∈ ℓ p ⊕ ∞ ℓ 2 . j Then { g v j } ∞ j = 1 is the unit vector basis of X p , v and { g w j } ∞ j = 1 is the unit vector basis of X p , w . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ g v j = e j + v j f j ∈ ℓ p ⊕ ∞ ℓ 2 and g w = e j + w j f j ∈ ℓ p ⊕ ∞ ℓ 2 . j Define also ∆ = ∆( w , v ) ∆ : X p , w → X p , v by ∆ g w = δ j g v j . j Note that ∆ is the restriction to X p , w of K : ℓ p ⊕ ∞ ℓ 2 → ℓ p ⊕ ∞ ℓ 2 defined by K ( e j ) = δ j e j and K ( f j ) = f j Consequently, � ∆ � ≤ � K � = max { 1 , max 1 ≤ j < ∞ δ j } . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ g v j = e j + v j f j ∈ ℓ p ⊕ ∞ ℓ 2 and g w = e j + w j f j ∈ ℓ p ⊕ ∞ ℓ 2 . j Define also ∆ = ∆( w , v ) ∆ : X p , w → X p , v by ∆ g w = δ j g v j . j Note that ∆ is the restriction to X p , w of K : ℓ p ⊕ ∞ ℓ 2 → ℓ p ⊕ ∞ ℓ 2 defined by K ( e j ) = δ j e j and K ( f j ) = f j Consequently, � ∆ � ≤ � K � = max { 1 , max 1 ≤ j < ∞ δ j } . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ g v j = e j + v j f j ∈ ℓ p ⊕ ∞ ℓ 2 and g w = e j + w j f j ∈ ℓ p ⊕ ∞ ℓ 2 . j Define also ∆ = ∆( w , v ) ∆ : X p , w → X p , v by ∆ g w = δ j g v j . j Note that ∆ is the restriction to X p , w of K : ℓ p ⊕ ∞ ℓ 2 → ℓ p ⊕ ∞ ℓ 2 defined by K ( e j ) = δ j e j and K ( f j ) = f j Consequently, � ∆ � ≤ � K � = max { 1 , max 1 ≤ j < ∞ δ j } . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ g v j = e j + v j f j ∈ ℓ p ⊕ ∞ ℓ 2 and g w = e j + w j f j ∈ ℓ p ⊕ ∞ ℓ 2 . j Define also ∆ = ∆( w , v ) ∆ : X p , w → X p , v by ∆ g w = δ j g v j . j Note that ∆ is the restriction to X p , w of K : ℓ p ⊕ ∞ ℓ 2 → ℓ p ⊕ ∞ ℓ 2 defined by K ( e j ) = δ j e j and K ( f j ) = f j Consequently, � ∆ � ≤ � K � = max { 1 , max 1 ≤ j < ∞ δ j } . Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ Denote by { h w j } the dual basis to { g w j } (and by { h v j } the dual basis to { g v j } , It was proved by Rosenthal that [ h w j ] and [ h v j ] contain copies of ℓ r for all q = p / ( p − 1 ) ≤ r ≤ 2 A major part in our proof is the fact that for any sequence r i ր 2 1 ri − 1 (i.e. d ( ℓ n i r i , ℓ n i 2 ր ∞ 2 ) → ∞ ) there are and n i such that n i sequences v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 such that δ j = w j / v j → 0 and ∆ ∗ isomorphically uniformly preserves these copies of ℓ n i r i . ( ∆ ∗ also preserves the modular space ℓ { r i } .) Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ Denote by { h w j } the dual basis to { g w j } (and by { h v j } the dual basis to { g v j } , It was proved by Rosenthal that [ h w j ] and [ h v j ] contain copies of ℓ r for all q = p / ( p − 1 ) ≤ r ≤ 2 A major part in our proof is the fact that for any sequence r i ր 2 1 ri − 1 (i.e. d ( ℓ n i r i , ℓ n i 2 ր ∞ 2 ) → ∞ ) there are and n i such that n i sequences v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 such that δ j = w j / v j → 0 and ∆ ∗ isomorphically uniformly preserves these copies of ℓ n i r i . ( ∆ ∗ also preserves the modular space ℓ { r i } .) Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ Denote by { h w j } the dual basis to { g w j } (and by { h v j } the dual basis to { g v j } , It was proved by Rosenthal that [ h w j ] and [ h v j ] contain copies of ℓ r for all q = p / ( p − 1 ) ≤ r ≤ 2 A major part in our proof is the fact that for any sequence r i ր 2 1 ri − 1 (i.e. d ( ℓ n i r i , ℓ n i 2 ր ∞ 2 ) → ∞ ) there are and n i such that n i sequences v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 such that δ j = w j / v j → 0 and ∆ ∗ isomorphically uniformly preserves these copies of ℓ n i r i . ( ∆ ∗ also preserves the modular space ℓ { r i } .) Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ Denote by { h w j } the dual basis to { g w j } (and by { h v j } the dual basis to { g v j } , It was proved by Rosenthal that [ h w j ] and [ h v j ] contain copies of ℓ r for all q = p / ( p − 1 ) ≤ r ≤ 2 A major part in our proof is the fact that for any sequence r i ր 2 1 ri − 1 (i.e. d ( ℓ n i r i , ℓ n i 2 ր ∞ 2 ) → ∞ ) there are and n i such that n i sequences v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 such that δ j = w j / v j → 0 and ∆ ∗ isomorphically uniformly preserves these copies of ℓ n i r i . ( ∆ ∗ also preserves the modular space ℓ { r i } .) Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ Denote by { h w j } the dual basis to { g w j } (and by { h v j } the dual basis to { g v j } , It was proved by Rosenthal that [ h w j ] and [ h v j ] contain copies of ℓ r for all q = p / ( p − 1 ) ≤ r ≤ 2 A major part in our proof is the fact that for any sequence r i ր 2 1 ri − 1 (i.e. d ( ℓ n i r i , ℓ n i 2 ր ∞ 2 ) → ∞ ) there are and n i such that n i sequences v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 such that δ j = w j / v j → 0 and ∆ ∗ isomorphically uniformly preserves these copies of ℓ n i r i . ( ∆ ∗ also preserves the modular space ℓ { r i } .) Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ Denote by { h w j } the dual basis to { g w j } (and by { h v j } the dual basis to { g v j } , It was proved by Rosenthal that [ h w j ] and [ h v j ] contain copies of ℓ r for all q = p / ( p − 1 ) ≤ r ≤ 2 A major part in our proof is the fact that for any sequence r i ր 2 1 ri − 1 (i.e. d ( ℓ n i r i , ℓ n i 2 ր ∞ 2 ) → ∞ ) there are and n i such that n i sequences v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 such that δ j = w j / v j → 0 and ∆ ∗ isomorphically uniformly preserves these copies of ℓ n i r i . ( ∆ ∗ also preserves the modular space ℓ { r i } .) Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ Denote by { h w j } the dual basis to { g w j } (and by { h v j } the dual basis to { g v j } , It was proved by Rosenthal that [ h w j ] and [ h v j ] contain copies of ℓ r for all q = p / ( p − 1 ) ≤ r ≤ 2 A major part in our proof is the fact that for any sequence r i ր 2 1 ri − 1 (i.e. d ( ℓ n i r i , ℓ n i 2 ր ∞ 2 ) → ∞ ) there are and n i such that n i sequences v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 such that δ j = w j / v j → 0 and ∆ ∗ isomorphically uniformly preserves these copies of ℓ n i r i . ( ∆ ∗ also preserves the modular space ℓ { r i } .) Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ Denote by { h w j } the dual basis to { g w j } (and by { h v j } the dual basis to { g v j } , It was proved by Rosenthal that [ h w j ] and [ h v j ] contain copies of ℓ r for all q = p / ( p − 1 ) ≤ r ≤ 2 A major part in our proof is the fact that for any sequence r i ր 2 1 ri − 1 (i.e. d ( ℓ n i r i , ℓ n i 2 ր ∞ 2 ) → ∞ ) there are and n i such that n i sequences v = { v j } ∞ j = 1 and w = { w j } ∞ j = 1 such that δ j = w j / v j → 0 and ∆ ∗ isomorphically uniformly preserves these copies of ℓ n i r i . ( ∆ ∗ also preserves the modular space ℓ { r i } .) Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ For 1 < p < 2, we construct new ideals of the form I ∆ ∗ ( w , v ) , that is the ideal of all operators factoring through ∆ ∗ ( w , v ) , for different sequences ( w , v ) = { w i , v i } . More precisely, we build a continuum C of different sequences ( w , v ) such that I ∆ ∗ ( w , v ) are all different. This already produces a continuum of different ideals. If A ⊂ C one can look at the closed ideal generated by { ∆ ∗ ( w , v ) } ( w , v ) ∈A We show moreover that (with the right choice of C ) if A � = B then the two closed ideal generated by A and B are different. This produces the required 2 2 ℵ 0 ideals. Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ For 1 < p < 2, we construct new ideals of the form I ∆ ∗ ( w , v ) , that is the ideal of all operators factoring through ∆ ∗ ( w , v ) , for different sequences ( w , v ) = { w i , v i } . More precisely, we build a continuum C of different sequences ( w , v ) such that I ∆ ∗ ( w , v ) are all different. This already produces a continuum of different ideals. If A ⊂ C one can look at the closed ideal generated by { ∆ ∗ ( w , v ) } ( w , v ) ∈A We show moreover that (with the right choice of C ) if A � = B then the two closed ideal generated by A and B are different. This produces the required 2 2 ℵ 0 ideals. Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ For 1 < p < 2, we construct new ideals of the form I ∆ ∗ ( w , v ) , that is the ideal of all operators factoring through ∆ ∗ ( w , v ) , for different sequences ( w , v ) = { w i , v i } . More precisely, we build a continuum C of different sequences ( w , v ) such that I ∆ ∗ ( w , v ) are all different. This already produces a continuum of different ideals. If A ⊂ C one can look at the closed ideal generated by { ∆ ∗ ( w , v ) } ( w , v ) ∈A We show moreover that (with the right choice of C ) if A � = B then the two closed ideal generated by A and B are different. This produces the required 2 2 ℵ 0 ideals. Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ For 1 < p < 2, we construct new ideals of the form I ∆ ∗ ( w , v ) , that is the ideal of all operators factoring through ∆ ∗ ( w , v ) , for different sequences ( w , v ) = { w i , v i } . More precisely, we build a continuum C of different sequences ( w , v ) such that I ∆ ∗ ( w , v ) are all different. This already produces a continuum of different ideals. If A ⊂ C one can look at the closed ideal generated by { ∆ ∗ ( w , v ) } ( w , v ) ∈A We show moreover that (with the right choice of C ) if A � = B then the two closed ideal generated by A and B are different. This produces the required 2 2 ℵ 0 ideals. Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ , main proposition For appropriate ( w , v ) the operator T = ∆ ∗ ( w , v ) has the following properties: X (in our case X ∗ p , v ) is a Banach space with a 1-unconditional basis { e i } (in our case { h v i } ) . T : X → X is a norm one operator satisfying: (a) For every M there is a finite dimensional subspace E of X such that d ( E ) > M and � Tx � ≥ 1 / 2 for all x ∈ E . and (b) For every m there is an n such that every m -dimensional subspace E of [ e i ] i ≥ n satisfies γ 2 ( T | E ) ≤ 2. We proved the following Proposition. Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ , main proposition For appropriate ( w , v ) the operator T = ∆ ∗ ( w , v ) has the following properties: X (in our case X ∗ p , v ) is a Banach space with a 1-unconditional basis { e i } (in our case { h v i } ) . T : X → X is a norm one operator satisfying: (a) For every M there is a finite dimensional subspace E of X such that d ( E ) > M and � Tx � ≥ 1 / 2 for all x ∈ E . and (b) For every m there is an n such that every m -dimensional subspace E of [ e i ] i ≥ n satisfies γ 2 ( T | E ) ≤ 2. We proved the following Proposition. Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ , main proposition For appropriate ( w , v ) the operator T = ∆ ∗ ( w , v ) has the following properties: X (in our case X ∗ p , v ) is a Banach space with a 1-unconditional basis { e i } (in our case { h v i } ) . T : X → X is a norm one operator satisfying: (a) For every M there is a finite dimensional subspace E of X such that d ( E ) > M and � Tx � ≥ 1 / 2 for all x ∈ E . and (b) For every m there is an n such that every m -dimensional subspace E of [ e i ] i ≥ n satisfies γ 2 ( T | E ) ≤ 2. We proved the following Proposition. Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ , main proposition For appropriate ( w , v ) the operator T = ∆ ∗ ( w , v ) has the following properties: X (in our case X ∗ p , v ) is a Banach space with a 1-unconditional basis { e i } (in our case { h v i } ) . T : X → X is a norm one operator satisfying: (a) For every M there is a finite dimensional subspace E of X such that d ( E ) > M and � Tx � ≥ 1 / 2 for all x ∈ E . and (b) For every m there is an n such that every m -dimensional subspace E of [ e i ] i ≥ n satisfies γ 2 ( T | E ) ≤ 2. We proved the following Proposition. Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ , main proposition For appropriate ( w , v ) the operator T = ∆ ∗ ( w , v ) has the following properties: X (in our case X ∗ p , v ) is a Banach space with a 1-unconditional basis { e i } (in our case { h v i } ) . T : X → X is a norm one operator satisfying: (a) For every M there is a finite dimensional subspace E of X such that d ( E ) > M and � Tx � ≥ 1 / 2 for all x ∈ E . and (b) For every m there is an n such that every m -dimensional subspace E of [ e i ] i ≥ n satisfies γ 2 ( T | E ) ≤ 2. We proved the following Proposition. Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ , main proposition Proposition Let T : X = [ e i ] → X satisfy (a) and (b). Then there exist a subsequence of N , 1 = p 1 < q 1 < p 2 < q 2 < . . . with the following properties: Denoting for each k, G k = [ e i ] q k i = p k . Let C be a continuum of subsequences of N each two of which has a finite intersection. For each α ∈ C , P α : X → [ G k ] k ∈ α denotes the natural basis projection and T α = TP α . If α 1 , . . . , α s ∈ C (possibly with repetitions) and α ∈ C \ { α 1 , . . . , α s } then for all A 1 , . . . , A s ∈ L ( X ) and all B 1 , . . . , B s ∈ L ( X ) s � � T α − A i T α i B i � ≥ 1 / 4 . i = 1 Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ , main proposition Proposition Let T : X = [ e i ] → X satisfy (a) and (b). Then there exist a subsequence of N , 1 = p 1 < q 1 < p 2 < q 2 < . . . with the following properties: Denoting for each k, G k = [ e i ] q k i = p k . Let C be a continuum of subsequences of N each two of which has a finite intersection. For each α ∈ C , P α : X → [ G k ] k ∈ α denotes the natural basis projection and T α = TP α . If α 1 , . . . , α s ∈ C (possibly with repetitions) and α ∈ C \ { α 1 , . . . , α s } then for all A 1 , . . . , A s ∈ L ( X ) and all B 1 , . . . , B s ∈ L ( X ) s � � T α − A i T α i B i � ≥ 1 / 4 . i = 1 Gideon Schechtman Ideals in L ( L p )
More Large Ideals in L ( L p ) , 1 < p � = 2 < ∞ , main proposition Proposition Let T : X = [ e i ] → X satisfy (a) and (b). Then there exist a subsequence of N , 1 = p 1 < q 1 < p 2 < q 2 < . . . with the following properties: Denoting for each k, G k = [ e i ] q k i = p k . Let C be a continuum of subsequences of N each two of which has a finite intersection. For each α ∈ C , P α : X → [ G k ] k ∈ α denotes the natural basis projection and T α = TP α . If α 1 , . . . , α s ∈ C (possibly with repetitions) and α ∈ C \ { α 1 , . . . , α s } then for all A 1 , . . . , A s ∈ L ( X ) and all B 1 , . . . , B s ∈ L ( X ) s � � T α − A i T α i B i � ≥ 1 / 4 . i = 1 Gideon Schechtman Ideals in L ( L p )
If I have more time If I have time left Gideon Schechtman Ideals in L ( L p )
back to small ideals in L ( L 1 ) Theorem. [JPS] There are at least 2 ℵ 0 small closed ideals in L ( L 1 ) . The new ideals are a familty ( I U q ) 2 < q < ∞ , where U q : ℓ 1 → L 1 {− 1 , 1 } N maps the unit vector basis of ℓ 1 to a carefully chosen Λ( q ) -set of characters. The following lemma is the heart of the proof. Lemma p Let 1 ≤ p < q < ∞ , { v 1 , . . . , v N } ⊂ L q , and let T : L 1 → L N 2 be 1 an operator. Suppose that C and ǫ satisfy max ǫ i = ± 1 � � N i = 1 ǫ i v i � q ≤ CN 1 / 2 , and 1 min 1 ≤ i ≤ N � Tv i � 1 ≥ ǫ . 2 q − p 2 q . Then � T � ≥ ( ǫ/ C ) N Gideon Schechtman Ideals in L ( L p )
back to small ideals in L ( L 1 ) Theorem. [JPS] There are at least 2 ℵ 0 small closed ideals in L ( L 1 ) . The new ideals are a familty ( I U q ) 2 < q < ∞ , where U q : ℓ 1 → L 1 {− 1 , 1 } N maps the unit vector basis of ℓ 1 to a carefully chosen Λ( q ) -set of characters. The following lemma is the heart of the proof. Lemma p Let 1 ≤ p < q < ∞ , { v 1 , . . . , v N } ⊂ L q , and let T : L 1 → L N 2 be 1 an operator. Suppose that C and ǫ satisfy max ǫ i = ± 1 � � N i = 1 ǫ i v i � q ≤ CN 1 / 2 , and 1 min 1 ≤ i ≤ N � Tv i � 1 ≥ ǫ . 2 q − p 2 q . Then � T � ≥ ( ǫ/ C ) N Gideon Schechtman Ideals in L ( L p )
back to small ideals in L ( L 1 ) Theorem. [JPS] There are at least 2 ℵ 0 small closed ideals in L ( L 1 ) . The new ideals are a familty ( I U q ) 2 < q < ∞ , where U q : ℓ 1 → L 1 {− 1 , 1 } N maps the unit vector basis of ℓ 1 to a carefully chosen Λ( q ) -set of characters. The following lemma is the heart of the proof. Lemma p Let 1 ≤ p < q < ∞ , { v 1 , . . . , v N } ⊂ L q , and let T : L 1 → L N 2 be 1 an operator. Suppose that C and ǫ satisfy max ǫ i = ± 1 � � N i = 1 ǫ i v i � q ≤ CN 1 / 2 , and 1 min 1 ≤ i ≤ N � Tv i � 1 ≥ ǫ . 2 q − p 2 q . Then � T � ≥ ( ǫ/ C ) N Gideon Schechtman Ideals in L ( L p )
back to small ideals in L ( L 1 ) p ) ∗ with | u ∗ i in L N p / 2 Proof: Take u ∗ = ( L N 2 i | ≡ 1 so that ∞ 1 � u ∗ i , Tv i � = � Tv i � 1 ≥ ǫ . Then � 1 N N � � � T ∗ u ∗ ( T ∗ u ∗ ǫ N ≤ i , v i � := i )( b ) v i ( b ) db 0 i = 1 i = 1 � 1 N � ≤ sup | ( T ∗ u ∗ i )( a ) v i ( b ) | db 0 a ∈ [ 0 , 1 ] i = 1 � 1 N � v i ( b ) T ∗ u ∗ =: � i � L ∞ [ 0 , 1 ] db 0 i = 1 � 1 N � ≤ � T � � v i ( b ) u ∗ i � L Np / 2 db 0 ∞ i = 1 � 1 N � � � 1 p i ( c ) v i ( b ) | q dc q db � u ∗ ≤ � T � N | 2 q p 2 ] 0 [ N i = 1 Gideon Schechtman Ideals in L ( L p )
back to small ideals in L ( L 1 ) p ) ∗ with | u ∗ i in L N p / 2 Proof: Take u ∗ = ( L N 2 i | ≡ 1 so that ∞ 1 � u ∗ i , Tv i � = � Tv i � 1 ≥ ǫ . Then � 1 N N � � � T ∗ u ∗ ( T ∗ u ∗ ǫ N ≤ i , v i � := i )( b ) v i ( b ) db 0 i = 1 i = 1 � 1 N � ≤ sup | ( T ∗ u ∗ i )( a ) v i ( b ) | db 0 a ∈ [ 0 , 1 ] i = 1 � 1 N � v i ( b ) T ∗ u ∗ =: � i � L ∞ [ 0 , 1 ] db 0 i = 1 � 1 N � ≤ � T � � v i ( b ) u ∗ i � L Np / 2 db 0 ∞ i = 1 � 1 N � � � 1 p i ( c ) v i ( b ) | q dc q db � u ∗ ≤ � T � N | 2 q p 2 ] 0 [ N i = 1 Gideon Schechtman Ideals in L ( L p )
back to small ideals in L ( L 1 ) p ) ∗ with | u ∗ i in L N p / 2 Proof: Take u ∗ = ( L N 2 i | ≡ 1 so that ∞ 1 � u ∗ i , Tv i � = � Tv i � 1 ≥ ǫ . Then � 1 N N � � � T ∗ u ∗ ( T ∗ u ∗ ǫ N ≤ i , v i � := i )( b ) v i ( b ) db 0 i = 1 i = 1 � 1 N � ≤ sup | ( T ∗ u ∗ i )( a ) v i ( b ) | db 0 a ∈ [ 0 , 1 ] i = 1 � 1 N � v i ( b ) T ∗ u ∗ =: � i � L ∞ [ 0 , 1 ] db 0 i = 1 � 1 N � ≤ � T � � v i ( b ) u ∗ i � L Np / 2 db 0 ∞ i = 1 � 1 N � � � 1 p i ( c ) v i ( b ) | q dc q db � u ∗ ≤ � T � N | 2 q p 2 ] 0 [ N i = 1 Gideon Schechtman Ideals in L ( L p )
back to small ideals in L ( L 1 ) p ) ∗ with | u ∗ i in L N p / 2 Proof: Take u ∗ = ( L N 2 i | ≡ 1 so that ∞ 1 � u ∗ i , Tv i � = � Tv i � 1 ≥ ǫ . Then � 1 N N � � � T ∗ u ∗ ( T ∗ u ∗ ǫ N ≤ i , v i � := i )( b ) v i ( b ) db 0 i = 1 i = 1 � 1 N � ≤ sup | ( T ∗ u ∗ i )( a ) v i ( b ) | db 0 a ∈ [ 0 , 1 ] i = 1 � 1 N � v i ( b ) T ∗ u ∗ =: � i � L ∞ [ 0 , 1 ] db 0 i = 1 � 1 N � ≤ � T � � v i ( b ) u ∗ i � L Np / 2 db 0 ∞ i = 1 � 1 N � � � 1 p i ( c ) v i ( b ) | q dc q db � u ∗ ≤ � T � N | 2 q p 2 ] 0 [ N i = 1 Gideon Schechtman Ideals in L ( L p )
back to small ideals in L ( L 1 ) p ) ∗ with | u ∗ i in L N p / 2 Proof: Take u ∗ = ( L N 2 i | ≡ 1 so that ∞ 1 � u ∗ i , Tv i � = � Tv i � 1 ≥ ǫ . Then � 1 N N � � � T ∗ u ∗ ( T ∗ u ∗ ǫ N ≤ i , v i � := i )( b ) v i ( b ) db 0 i = 1 i = 1 � 1 N � ≤ sup | ( T ∗ u ∗ i )( a ) v i ( b ) | db 0 a ∈ [ 0 , 1 ] i = 1 � 1 N � v i ( b ) T ∗ u ∗ =: � i � L ∞ [ 0 , 1 ] db 0 i = 1 � 1 N � ≤ � T � � v i ( b ) u ∗ i � L Np / 2 db 0 ∞ i = 1 � 1 N � � � 1 p i ( c ) v i ( b ) | q dc q db � u ∗ ≤ � T � N | 2 q p 2 ] 0 [ N i = 1 Gideon Schechtman Ideals in L ( L p )
back to small ideals in L ( L 1 ) � 1 N N � � ǫ N ≤ � T ∗ u ∗ i , v i � := ( T ∗ u ∗ i )( b ) v i ( b ) db 0 i = 1 i = 1 ≤ ..... � 1 N � � � 1 p i ( c ) v i ( b ) | q dc q db � u ∗ ≤ � T � N | 2 q p 2 ] 0 [ N i = 1 � 1 N 2 q � � p i ( c ) v i ( b ) | q db dc � 1 � ≤ � T � N | u ∗ q p 2 ] [ N 0 i = 1 p + q 2 q . ≤ C � T � N So, � T � ≥ ( ǫ/ C ) N 1 − p + q q − p 2 q = ( ǫ/ C ) N 2 q . Gideon Schechtman Ideals in L ( L p )
back to small ideals in L ( L 1 ) � 1 N N � � ǫ N ≤ � T ∗ u ∗ i , v i � := ( T ∗ u ∗ i )( b ) v i ( b ) db 0 i = 1 i = 1 ≤ ..... � 1 N � � � 1 p i ( c ) v i ( b ) | q dc q db � u ∗ ≤ � T � N | 2 q p 2 ] 0 [ N i = 1 � 1 N 2 q � � p i ( c ) v i ( b ) | q db dc � 1 � ≤ � T � N | u ∗ q p 2 ] [ N 0 i = 1 p + q 2 q . ≤ C � T � N So, � T � ≥ ( ǫ/ C ) N 1 − p + q q − p 2 q = ( ǫ/ C ) N 2 q . Gideon Schechtman Ideals in L ( L p )
back to small ideals in L ( L 1 ) � 1 N N � � ǫ N ≤ � T ∗ u ∗ i , v i � := ( T ∗ u ∗ i )( b ) v i ( b ) db 0 i = 1 i = 1 ≤ ..... � 1 N � � � 1 p i ( c ) v i ( b ) | q dc q db � u ∗ ≤ � T � N | 2 q p 2 ] 0 [ N i = 1 � 1 N 2 q � � p i ( c ) v i ( b ) | q db dc � 1 � ≤ � T � N | u ∗ q p 2 ] [ N 0 i = 1 p + q 2 q . ≤ C � T � N So, � T � ≥ ( ǫ/ C ) N 1 − p + q q − p 2 q = ( ǫ/ C ) N 2 q . Gideon Schechtman Ideals in L ( L p )
back to small ideals in L ( L 1 ) � 1 N N � � ǫ N ≤ � T ∗ u ∗ i , v i � := ( T ∗ u ∗ i )( b ) v i ( b ) db 0 i = 1 i = 1 ≤ ..... � 1 N � � � 1 p i ( c ) v i ( b ) | q dc q db � u ∗ ≤ � T � N | 2 q p 2 ] 0 [ N i = 1 � 1 N 2 q � � p i ( c ) v i ( b ) | q db dc � 1 � ≤ � T � N | u ∗ q p 2 ] [ N 0 i = 1 p + q 2 q . ≤ C � T � N So, � T � ≥ ( ǫ/ C ) N 1 − p + q q − p 2 q = ( ǫ/ C ) N 2 q . Gideon Schechtman Ideals in L ( L p )
back to small ideals in L ( L 1 ) � 1 N N � � ǫ N ≤ � T ∗ u ∗ i , v i � := ( T ∗ u ∗ i )( b ) v i ( b ) db 0 i = 1 i = 1 ≤ ..... � 1 N � � � 1 p i ( c ) v i ( b ) | q dc q db � u ∗ ≤ � T � N | 2 q p 2 ] 0 [ N i = 1 � 1 N 2 q � � p i ( c ) v i ( b ) | q db dc � 1 � ≤ � T � N | u ∗ q p 2 ] [ N 0 i = 1 p + q 2 q . ≤ C � T � N So, � T � ≥ ( ǫ/ C ) N 1 − p + q q − p 2 q = ( ǫ/ C ) N 2 q . Gideon Schechtman Ideals in L ( L p )
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