The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Laustsen, Niels Jakob; Loy, Richard J.; Read, C. J.

doi:10.1016/j.jfa.2004.02.009

Cited by 34 publications

(34 citation statements)

References 24 publications

(21 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…For a sequence of operators U m : (⊕ m i=1 H i ) ℓ ∞ (m) → K, where H i and K are Hilbert spaces, [7,Lemma 5.3] leads to a dichotomy theorem analogous to Corollary 7. As a result, one may adapt the foregoing arguments to prove that for E = (⊕ℓ 2 (n)) ℓ 1 , G ℓ 1 (E)) is the unique maximal ideal of L(E).…”

Section: Theorem 5 Suppose That T ∈ L(e) and Condition ( * ) Holds Letmentioning

confidence: 99%

“…In this note, we make a small contribution to the program by identifying the unique maximal ideal in the algebra L(E), where E = (⊕ℓ ∞ (n)) ℓ 1 . Our method is slightly more general in the sense that it works also for the space E = (⊕ℓ 2 (n)) ℓ 1 , which gives an alternate proof of the main result of [9]. Let E n be finite dimensional Banach spaces and let E = (⊕E n ) ℓ 1 .…”

mentioning

confidence: 99%

“…The unique maximal ideal in the Banach algebra L(E), E = (⊕ℓ ∞ (n)) ℓ 1 , is identified. The proof relies on techniques developed by Laustsen, Loy and Read [7] and a dichotomy result for operators mapping into L 1 due to Laustsen, Odell, Schlumprecht and Zsák [8].Given a Banach space E, it is a natural problem to try to understand the ideal structure of the Banach algebra L(E). In the classical period, the only Banach spaces E for which the lattice of closed ideals in L(E) are fully understood are Hilbert space [2,4,11] and the sequence spaces c 0 and ℓ p , 1 ≤ p < ∞ [3].…”

mentioning

confidence: 99%

“…In the classical period, the only Banach spaces E for which the lattice of closed ideals in L(E) are fully understood are Hilbert space [2,4,11] and the sequence spaces c 0 and ℓ p , 1 ≤ p < ∞ [3]. In the past decade, starting with [7], there has been a resurgence of interest in the problem and new results have been obtained, see, e.g., [5,6,8,9,10,12,13]. One should also mention the recent breakthrough example AH by Argyros and Haydon [1], where L(AH) is well understood because of a very different reason.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 4 more Smart Citations

Ideals of operators on $(\oplus \ell ^\infty (n))_{\ell ^1}$

Leung

2015

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. The unique maximal ideal in the Banach algebra L(E), E = (⊕ℓ ∞ (n)) ℓ 1 , is identified. The proof relies on techniques developed by Laustsen, Loy and Read [7] and a dichotomy result for operators mapping into L 1 due to Laustsen, Odell, Schlumprecht and Zsák [8].Given a Banach space E, it is a natural problem to try to understand the ideal structure of the Banach algebra L(E). In the classical period, the only Banach spaces E for which the lattice of closed ideals in L(E) are fully understood are Hilbert space [2,4,11] and the sequence spaces c 0 and ℓ p , 1 ≤ p < ∞ [3]. In the past decade, starting with [7], there has been a resurgence of interest in the problem and new results have been obtained, see, e.g., [5,6,8,9,10,12,13]. One should also mention the recent breakthrough example AH by Argyros and Haydon [1], where L(AH) is well understood because of a very different reason. In this note, we make a small contribution to the program by identifying the unique maximal ideal in the algebra L(E), where E = (⊕ℓ ∞ (n)) ℓ 1 . Our method is slightly more general in the sense that it works also for the space E = (⊕ℓ 2 (n)) ℓ 1 , which gives an alternate proof of the main result of [9]. Let E n be finite dimensional Banach spaces and let E = (⊕E n ) ℓ 1 . Given a subset I of N, denote by E I the subspace (⊕ n∈I E n ) ℓ 1 . The natural embedding of E I into E is denoted by J I and the natural projection from E onto E I is denoted by P I . If I = {n}, then we write for short J n and P n respectively. Let T be a bounded linear operator on E, then T has a natural matrix representation (T mn ), where T mn = P m T J n : E n → E m . Denote the nth column in this representation by T n . Thus T n = T J n :Let X, Y, Z be Banach spaces. If T : X → Y is a bounded linear operator and ε > 0, defineFor Z = c 0 , respectively ℓ 1 , we write Fac In the sequel, T will always denote a bounded linear operator on E.

show abstract

Section: Theorem 5 Suppose That T ∈ L(e) and Condition ( * ) Holds Letmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Ideals of operators on $(\oplus \ell ^\infty (n))_{\ell ^1}$

Leung

2015

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Structure of modules over the stereotype algebra ℒ(X) of operators

Akbarov¹

2006

Funct Anal Its Appl

View full text Add to dashboard Cite

It is well known that every module M over the algebra L (X) of operators on a finitedimensional space X can be represented as the tensor product of X by some vector space E , M ∼ = E ⊗ X . We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the stereotype algebra L (X) of operators on X there exists a unique (up to isomorphism) stereotype space E such that M lies between two natural stereotype tensor products of E by X ,As a corollary, we show that if X is a nuclear Fréchet space with a basis, then each Fréchet module M over the stereotype operator algebra L (X) can be uniquely represented as the projective tensor product of X by some Fréchet space E , M = E ⊗ X . Statement of the Problem and Main ResultsOne of the classical results in the theory of associative algebras (e.g., see [1]) states that every module M over the algebra L (X) of operators on a finite-dimensional space X (say, over C) can be represented as the direct sum m X (where m is a cardinal number depending on M ) of a family {X i ; i ∈ m} of copies of X , M = m X.This result cannot be generalized to the case of infinite-dimensional spaces X , since otherwise the algebra L (X) treated as a left module over itself would also have the form L (X) = m X and hence L (X) would consist only of finite-dimensional operators. Indeed, suppose that there exists an isomorphism ι : m X → L (X). Let i 0 ∈ m, and let {x i ; i ∈ m} ∈ m X be a family all of whose elements but the i 0 th are zero,Then the image ι({x i }) of this family is invariant under one-dimensional operators of the form g ⊗ a, where g(a) = 1,and hence is a one-dimensional operator in L (X). Since every family {x i ; i ∈ m} ∈ m X is a finite sum of families with only one nonzero element, we see that the image ι({x i }) of an arbitrary family {x i ; i ∈ m} ∈ m X must be a finite-dimensional operator; i.e., every operator in L (X) is finite-dimensional. This is obviously impossible if X is infinite-dimensional, and we arrive at a contradiction.If we think of the direct sum m X as the tensor product of X by some other vector space E (of algebraic dimension m), then we can state the preceding as follows.

show abstract