The lattice of closed ideals in the Banach algebra of operators on a certain dual Banach space

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1 The lattice of closed ideals in the Banach algebra of operators on a certain dual Banach space Niels Jakob Laustsen, Thomas Schlumprecht, and András Zsák Abstract In this paper we determine the closed operator ideals of the space F := ( n N ln Key words: Ideal lattice, operator, Banach space, Banach algebra. 000 Mathematics Subject Classication: primary 47L0, 46H0; secondary 47L0, 46B45. l. Introduction The aim of this note is to classify the closed ideals in the Banach algebra B(F of (bounded, linear operators on the Banach space ( F := l n. (. l n N More precisely, we shall show that there are exactly four closed ideals in B(F, namely {0}, the compact operators K (F, the closure G l (F of the set of operators factoring through l, and B(F itself. The collection of Banach spaces E for which a classication of the closed ideals in B(E exists is very sparse. Indeed, the following list appears to be the complete list of such spaces. (i For a nite-dimensional Banach space E, B(E = M n, where n is the dimension of E, and so it is ancient folklore that B(E is simple in this case. (ii In 94 Calkin [] classied all the ideals in B(l. In particular he proved that there are only three closed ideals in B(l, namely {0}, K (l, and B(l. (iii In 960 Gohberg, Markus, and Feldman [5] extended Calkin's theorem to the other classical sequence spaces. More precisely, they showed that {0}, K (E, and B(E are the only closed ideals in B(E for each of the spaces E = c 0 and E = l p, where p <.

2 (iv Later in the 960'ies Gramsch [6] and Luft [9] independently extended Calkin's theorem in a dierent direction by classifying all the closed ideals in B(H for each Hilbert space H (not necessarily separable. In particular, they showed that these ideals are well-ordered by inclusion. (v In 003 Laustsen, Loy, and Read [7] proved that, for the Banach space ( E := l n, (. c 0 n N there are exactly four closed ideals in B(E, namely {0}, the compact operators K (E, the closure G c0 (E of the set of operators factoring through c 0, and B(E itself. Note that the Banach space F given by (. is the dual of the Banach space E given by (., and so the result of this note can be seen as a `dualization' of [7]. In fact, our strategy draws heavily on the methods introduced in [7]. The classication theorem We begin this section by recalling various denitions and results from [7]. For simplicity we state the results only in the generality that is required for our present purposes, but emphasize that a number of them hold true in greater generality.. l -direct sums. Let (E n be a sequence of Banach spaces. We denote by ( E n l the l -direct sum of E, E,..., that is, the collection of sequences (x n such that x n E n for each n N and (xn := x n <. (. n= This is a Banach space for coordinatewise dened addition and scalar multiplication and norm given by (.. Set E := ( E n l. For each m N, we write J E m for the canonical embedding of E m into E and Q E m for the canonical projection of E onto E m. Both J E m and Q E m are operators of norm one; in fact, the former is an isometry, and the latter is a quotient map. When no ambiguity may arise, we omit the superscript E from the operators J E m and Q E m. We use similar notation and conventions for nite collections of Banach spaces and operators.. Denition. Let (E n and (F n be sequences of Banach spaces, and let T : ( E n l ( Fn be an operator. We associate with T the innite matrix (T m,n, where l T m,n := Q F mt J E n : E n F m (m, n N. The support of the n th column of T is colsupp n (T := { m N T m,n 0 } (n N.

3 We say that T has nite columns if each column has nite support, and in this case we set µ n (T := max ( colsupp n (T. The signicance of operators with nite columns lies in the fact that, in the case where each of the spaces E n (n N is nite-dimensional, for each operator T : ( E n l ( Fn l we can nd a perturbation T : ( E n l ( F n with nite columns such that the dierence T T is compact and has arbitrarily small norm (see [7, Lemma.7(i]..3 Diagonal operators. Let (E n and (F n be sequences of Banach spaces, and, for each n N, let T n : E n F n be an operator. Suppose that sup T n <. Then we can dene the diagonal operator ( ( diag(t n : (x n (T n x n, En Fn Clearly, we have diag(tn = sup Tn. l The following construction is a dual version of [7, Construction 4.]..4 Construction. Let (E n and (F n be sequences of Banach spaces, and set E := ( En l and F := ( F n l. Further, set F := ( Fn l, where F n := F for each n N. Let T : E F be an operator. Since T Jn E T for each n N, we have a diagonal operator diag(t Jn E : E F. Dene ( W : F F, y = (y n n N Q F my n. Then it follows that W and T can be written as n= l m N T = W diag(t J E n. (..5 Denition. Let G be a closed subspace of a Hilbert space H. We denote by G the orthogonal complement of G in H, and write proj G for the orthogonal projection of H onto G (so that proj G is the idempotent operator on H with image G and kernel G. Let m N, let E be a Banach space, and let K,..., K m be Hilbert spaces. For each operator T : E (K K m l and each ε 0, set (proj n ε (T := sup n N G proj G m T > ε 0 whenever G j K j are subspaces N 0 {± }. with dim G j n, for j =,..., m.6 Lemma. Let m N, let H and K,..., K m be Hilbert spaces, let T : H (K K m l be an operator, and let 0 < ε < T. (i Suppose that n ε (T is nite. Then there are operators R: H l and S : l (K K m l such that T SR ε, R T n ε (T +, and S. 3 l.

4 (ii For each natural number n n ε (T / +, there are operators U : l n H and V : (K K m l l n such that I l n = V T U, U /ε, and V. (iii Let k N, let H 0 be a closed conite subspace of H, and suppose that n ε (T dim H0 + k. Then n ε (T H0 k. The proofs of (i and (ii of Lemma.6 follow along the same line as the proof of [7, Lemma 5.3(i(ii]. Actually we could deduce (i and (ii by applying [7, Lemma 5.3(i (ii] to some variant of the adjoint of T. For the sake of being selfcontained we include a proof. Proof. (i. Let n = n ε + and choose subspaces G j K j, dim G j n, for j =,... m, so that m (proj G proj G... proj G m T = proj G j Q j T ε. We put d = m j= dim G j. For j =,... m we let I j : G j l dim G j be an isometry and Ĩ j : l dim G j l dim G j the formal identity. For x H we dene R(x = ( Ĩ j I j proj Gj Q j T (x m ( m j= j= l dim G j and for z = (z j m j= ( m j= l dim G j l we dene S(z = ( I j j= Ĩ j (z j m ( m j= j= K j l. l = l d, Since Ĩj dim G j n and Ĩ j it follows that R T n and S. Furthermore, T SR = (proj G proj G... proj G m T, which proves (i. (ii. Let n nε(t + By induction we will choose normalized elements x, x,... x n H so that a x i x i, whenever i i are in {,... n}, b T (x i T (x i, whenever i i are in {,... n} and j {,... m}, c T (x i = m j= Q jt (x i > ε, whenever i {,... n}. Choose x H, x = so that T (x ε, and assuming that x, x,..., x k were chosen for some k n, we put for j =,..., m G j = span{q j T (x i, Q j T T T x i : i =,... k } K j. Since dim G j k n n ε (T, there is a w H, w =, so that (proj G... proj G m T (w > ε. We write w = k α i x i + βt T x i + z, with z span {x i, T T x i : i =,... k}. i= 4

5 Since proj G j Q j T (x i = proj G j Q j T T T x i = 0, it follows that (proj G...., proj G m T (w = (proj G...., proj G m T (z and, thus, x k = z/ z satises the conditions (a, (b and (c. Finally we dene U : l n H, e i x i, for i =,... n ε and V : ( m n m y j, Q j T (x i ε j= K j l l k, (y j j= e i Q j T (x i m l= Q lt (x i. j= Because of (a we have U /ε, and using the triangle inequality and (c it is easy to see that V and using (b we deduce that I l n = V T U. (iii. For each j =,..., m, let G j be a subspace of K j with dim G j k. Set F j := G j + Q j T (H0. Then F j is nite-dimensional with dim F j n ε (T, and so we can nd x H such that x and (projf proj F m T x > ε. It follows that (projg proj G m T H0 (projg proj G m T (proj H0 x i= (proj F proj F m T (proj H0 x = (projf proj F m T x > ε, and so n ε (T H0 k..7 ( Remark. Let (K n be a sequence of Hilbert spaces, and let T be an operator on Kn l with nite columns. As in [7, Remark 5.4], there is a natural way to dene n ε (T J m for each ε 0 and each m N, namely by ignoring the conite number of Hilbert spaces K k such that Q k T J m = 0. For each pair (E, F of Banach spaces, set G l (E, F := { T S S B(E, l, T B(l, F }. The fact that l = l l implies that G l is an operator ideal, and so its closure G l is a closed operator ideal. As usual, we write G l (E instead of G l (E, E.8 Lemma. Let E be a Banach space and J be an ideal in B(E. If P is an idempotent operator on E and P J, then in fact P J. Proof. Let (T n be a sequence in J converging to P. Replacing T n with P T n P we may assume that T n P B(EP for all n N. Note that P B(EP is a Banach algebra with unit P, and so there exists n such that T n is invertible. Thus there is an operator U B(E with P = (P UP T n, which implies that P J..9 Theorem. Set F := ( l n. l following conditions are equivalent: For each operator T on F with nite columns the 5

6 (i T G l (F, (ii sup { n ε (T J k k N } = for some ε > 0, (iii there are operators U and V on F such that V T U = I F. Proof. We begin by proving the implication not (ii not (i Let 0 < ε < T, and suppose that c := sup { n ε (T J k k N } <. Then, for each k N, there are operators R k : l k l and S k : l F such that T J k S k R k ε, R k T c +, and S k by Lemma.6(i. In the notation of Construction.4 (with E n = F n = l n, we see that the diagonal operators diag(r k : F ( l l and diag(s k : ( l l F (where F = ( n=f l satisfy diag(t Jk diag(s k diag(r k = sup T J k S k R k ε. It follows that diag(t J k G l (F, F because ( l l is isomorphic to l and ε is arbitrary, and so by (. we conclude that T G l (F, as desired. To show (ii (iii suppose that sup { n ε (T J k k N } = for some ε > 0. We construct inductively a strictly increasing sequence (k j of natural numbers such that, for each j N, the following assertions hold: (a colsupp kj (T and µ kj+ (T µ kj (T. (b Set m 0 := 0, m j := µ kj (T, and E j := ( m j i=m j + li l, and let P j : F E j be the canonical projection. Then there are operators U j : l j l k j and V j : E j l j such that the diagram I l j U j l k j J kj is commutative, U j /ε, V j, and im U j condition is ignored for j =. F T F P j l j E j V j m j i= ker T i,kj. (The latter We start the induction by choosing k N such that n ε (T J k. Then colsupp k (T and T J k > ε. Take a unit vector x l k such that T J k x > ε, and dene U : α α n i= x T J k x, l l k. Further, take a functional V : E l of norm such that V (P T J k x = P T J k. Then the diagram in (b is commutative because P T J k x = T J k x. Now let j, and suppose that k < k < < k j have been chosen in accordance with (a(b. Set h := m j i= i, take k j > k j such that n ε (T J kj h + (j, 6

7 and set H := m j i= ker T i,kj = ker( m j i= Q T J kj. Since dim H k j h it follows that dim H h and hence n ε (T J kj H (j by Lemma.6(iii. In particular T J kj H 0, and (a is satised. Further, we note that n ε (P j T J kj H = n ε (T J kj H because Q i T J kj H = 0 whenever i m j or i > m j. Lemma.6(ii then implies that there are operators U j : l j H l k j and V j : E j l j such that (b is satised. This completes the inductive construction. We `glue' the operators U j (j N together in the following way to obtain an operator U on F. Given x F, dene y i l i by { U j Q j x if i = k j for some j N y i := (i N. 0 otherwise Then we have y i = i= j= U j Q j x x ε <, and so Ux := (y i denes an operator U on F. Similarly, since V j P j x j= P j x = x, j= the assignment V x := (V j P j x denes an operator V on F. We claim that V T U = I F. To this end, it suces to check that { x if i = j Q i V T UJ j x = (i, j N, x l j. (.3 0 otherwise By denition, we have Q i V T UJ j x = V i P i T J kj U j x. For i = j, the diagram in (b, above, shows that this is x. For i < j, we have U j x ker T h,kj ( h m j, and so P i T J kj U j = m i h=m i + J h Q h T J kj U j x = m i h=m i + J h T h,kj U j x = 0. For i > j, P i T J kj = m i h=m i + J ht h,kj = 0 because T h,kj = 0 whenever h m j. This completes the proof of the implication (ii (iii. Finally to deduce (iii (i we need to observe that G l (F is a proper ideal in B(F, and, thus, cannot contain T, since by (iii I F is contained in the ideal generated by T. First note that by Lemma.8, if I F G l (F, then I F G l (F, and hence F would be isomorphic to l. It is known, however, that F is not isomorphic to l, although this is by no means obvious. One may for example use that l has a unique unconditional basis up to equivalence (a fact that essentially relies on Khintchine's inequality, whereas it is easy to see that F does not have this property. Our main theorem on the classication of the closed ideals of B(F can be deduced. 7

8 .0 Theorem. Set F := ( l n l. The lattice of closed ideals in B(F is given by {0} K (F G l (F B(F. (.4 Proof. It is clear that {0} K (F G l (F (F contains l as a complemented subspace, the projection onto which is an example of a non-compact operator in G l (F. By (ii (i in Theorem.9 the identity on F cannot be in G l (F, thus G l (F is a proper subideal of B(F. We now show that the ideals in (.4 are the only closed ideals of B(F. Standard basis arguments show that the identity on l factors through any non-compact operator in B(F (see for example [7, Ÿ3]. It follows that for each non-zero, closed ideal J in B(F, either J = K (F or G l (F J. Suppose that J is a closed ideal in B(F properly containing G l (F. Take T J \ G l (F, and take T B(F with nite columns such that T T is compact (cf. [7, Lemma.7(i]. Then T is also in J \G l (F. By Theorem.9 we conclude that I F = V T U for some operators U and V on F. It follows that J = B(F, as required. In [, Ÿ8] Bourgain, Casazza, Lindenstrauss, and Tzafriri prove that every innitedimensional, complemented subspace of the Banach space F := ( l n l is isomorphic to either F or l. Here we present a new proof of this fact using only the ideal structure of B(F. More precisely, we shall deduce it from the dichotomy in Theorem.0 for operators in B(F with nite columns.. Remark. In [7, Ÿ6] a new proof is presented for the corresponding result of Bourgain, Casazza, Lindenstrauss, and Tzafriri for the space E := ( l n c 0, which says that every innite-dimensional, complemented subspace of E is isomorphic to either E or c 0. This new proof in [7] relies on a result of Casazza, Kottman and Lin [3] that implies that E is primary. The result of [3], however, does not show that F is primary, and so the argument in [7] cannot be used here. The proof we present below uses only the classication result, Theorem.0, and it also works for the space E. We start with an easy strenghtening of part (ii of Theorem.0.. Proposition. Let T be an operator on F. If T G l (F then there exist operators A and B on F such that I F = AT B. Proof. Let K be a compact operator on F such that T K has nite columns. Note that by the ideal property we have T K G l (F. By Therorem.0 there are operators U and V on F such that I F = U(T KV. Thus UT V is a compact perturbation of the identity, and hence it is a Fredholm operator. It follows that for some W B(F the operator W U T V is a conite-rank projection. Since E is isomorphic to its nite-codimensional subspaces the result follows..3 Theorem. (Bourgain, Casazza, Lindenstrauss, Tzafriri [] Every innite-dimensional, complemented subspace of F = ( l n is isomorphic to either l F or l. 8

9 Proof. Let Y be an innite-dimensional, complemented subspace of F, and let P B(F be an idempotent operator with image Y. If P G l (F, then by Lemma.8 we have P G l (F, and hence Y is isomorphic to l. If P G l (F, then by Propostion. the identity on F factors through P, i.e., F is isomorphic to a complemented subspace of Y. We can thus write F Y V and Y F W for suitable Banach spaces V and W. We now use Pelczynski's decomposition method to show that Y is isomorphic to F. F Y V F W V (F F... l W V (Y V Y V... l W V (Y V Y V... l W F W Y, where we also used the fact that F is isomorphic to (F F... l. Acknowledgements This paper was initiated during a visit of the rst author to Texas A&M University. He acknowledges with thanks the nancial support from the Danish Natural Science Research Council and NSF Grant number DMS that made this visit possible. He also wishes to thank his hosts for their very kind hospitality during his stay. References [] J. Bourgain, P. G. Casazza, J. Lindenstrauss, and L. Tzafriri, Banach spaces with a unique unconditional basis, up to permutation, Mem. American Math. Soc. 3, 985. [] J. W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Annals of Math. 4 (94, [3] P. G. Casazza, C. A. Kottman, Bor Luh Lin, On some classes of primary Banach spaces, Canad. J. Math. 9 (977, no. 4, [4] J. Diestel, A survey of results related to the DunfordPettis property, in Proceedings of the conference on integration, topology, and geometry in linear spaces (ed. W. H. Graves, Contemp. Math., American Math. Soc., Providence, R. I., 980. [5] I. C. Gohberg, A. S. Markus, and I. A. Feldman, Normally solvable operators and ideals associated with them, American Math. Soc. Translat. 6 (967, 6384, Russian original in Bul. Akad. tiince RSS Moldoven 0 (76 (960,

10 [6] B. Gramsch, Eine Idealstruktur Banachscher Operatoralgebren, J. Reine Angew. Math. 5 (967, 975. [7] N. J. Laustsen, R. J. Loy, and C. J. Read, The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces, J. Functional Anal., to appear. [8] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, Vol. I, Ergeb. Math. Grenzgeb. 9, Springer-Verlag, 977. [9] E. Luft, The two-sided closed ideals of the algebra of bounded linear operators of a Hilbert space, Czechoslovak Math. J. 8 (968, [0] A. Pietsch, Operator ideals, North-Holland, 980. N. J. Laustsen, Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK00 Copenhagen Ø, Denmark; laustsen@math.ku.dk. T. Schlumprecht, Department of Mathematics, Texas A&M University, College Station, TX 77843, USA; schlump@math.tamu.edu. A. Zsák, Department of Mathematics, Texas A&M University, College Station, TX 77843, USA; azsak@math.tamu.edu and Fitzwilliam College, Cambridge CB3 0DG, England 0