Maximal Ideals in the Algebra of Operators on Certain Banach Spaces

Laustsen, Niels Jakob

doi:10.1017/s0013091500001097

Cited by 26 publications

(28 citation statements)

References 30 publications

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“…: Suppose that PAG F ðEÞ: Then in fact PAG F ðEÞ by [22,Proposition 3.4], and so P ¼ P n j¼1 S j R j for some nAN; R 1 ; y; R n ABðE; F Þ; and S 1 ; y; S n ABðF ; EÞ: Clearly, the operators R : x/ðR 1 x; y; R n xÞ; E-F "n ; and S : ðx 1 ; y; x n Þ/ P n j¼1 S j x j ; F "n -E; satisfy P ¼ SR: This implies by Laustsen [22, Lemma 3.6(ii)] that Q :¼ RSRSABðF "n Þ is idempotent with im QDim P:…”

Section: Operators On C 0 -Direct Sumsmentioning

confidence: 99%

“…Laustsen has proved that this maximal ideal is the only maximal ideal in BðJÞ; and applied this result to construct Banach spaces E such that BðEÞ has any specified finite number of maximal ideals of any specified codimensions (see [22]). …”

Section: Introductionmentioning

confidence: 99%

“…Fourth, while solving the unconditional basic sequence problem, Gowers and Maurey constructed the first example of a hereditarily indecomposable Banach space, and showed that the ideal SðEÞ of strictly singular operators is a maximal ideal of codimension one in BðEÞ for each such space E (see [14]); once again this maximal ideal is unique (see [22]). Androulakis and Schlumprecht have proved that non-compact, strictly singular operators exist on the particular hereditarily indecomposable Banach space E that Gowers and Maurey constructed (see [1]), and so in this case SðEÞ is not the only non-trivial, closed ideal in BðEÞ: It is a major open problem whether or not there exists a Banach space E such that the ideal of compact operators is a maximal ideal of codimension one in BðEÞ: The reader is referred to Schlumprecht's paper [28] for the current state of this difficult problem together with an impressive new method of attack.…”

Section: Introductionmentioning

confidence: 99%

“…Later, when solving Banach's hyperplane problem, Gowers constructed a Banach space G such that c N =c 0 is a quotient of BðGÞ (see [13,15]). Laustsen has classified the maximal ideals in BðGÞ by observing that each such ideal is the preimage of a maximal ideal in c N =c 0 (see [22]). …”

Section: Introductionmentioning

confidence: 99%

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

Laustsen¹,

Loy

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2004

Journal of Functional Analysis

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Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra BðEÞ of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c 0 and c p for 1ppoN: We add a new member to this family by showing that there are exactly four closed ideals in BðEÞ for the Banach space E :¼ ð"c n 2 Þ c0 ; that is, E is the c 0 -direct sum of the finite-dimensional Hilbert spaces c 1 2 ; c 2 2 ; y; c n 2 ; y . r 2004 Elsevier Inc. All rights reserved. MSC: primary 47L10; 46H10; secondary 47L20; 46B45

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Section: Operators On C 0 -Direct Sumsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Laustsen¹,

Loy

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2004

Journal of Functional Analysis

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“…Laustsen [25,26] has subsequently shown that W (J 2 ) is the unique maximal ideal in B(J 2 ), while the scalar multiples of the character are the only traces on B(J 2 ). (vi) Giesy and James [20] have shown that c 0 is finitely representable in J 2 , that is, for each ε > 0 and each n ∈ N, there is an operator T :…”

mentioning

confidence: 99%

An amalgamation of the Banach spaces associated with James and Schreier, Part I: Banach-space structure

Bird¹,

Laustsen²

2010

Banach Center Publ.

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Abstract. We create a new family of Banach spaces, the James-Schreier spaces, by amalgamating two important classical Banach spaces: James' quasi-reflexive Banach space on the one hand and Schreier's Banach space giving a counterexample to the Banach-Saks property on the other. We then investigate the properties of these James-Schreier spaces, paying particular attention to how key properties of their 'ancestors' (that is, the James space and the Schreier space) are expressed in them. Our main results include that each James-Schreier space is c0-saturated and that no James-Schreier space embeds in a Banach space with an unconditional basis.1. Introduction. The purpose of this paper is to introduce a new family of Banach spaces which we call the James-Schreier spaces because they arise by amalgamating the definitions of the quasi-reflexive James spaces with the Schreier space. The original motivation behind these spaces was to produce a new example of a Banach sequence algebra with a bounded approximate identity, in analogy with Andrew and Green's study of the James space as a Banach algebra [3]. This idea turned out to be successful, as essentially all results about the James space as a Banach algebra carry over to our new spaces; see [9] for details.Having thus reached our initial goal, we soon realized that a serious problem was lurking in the background, namely: how can we distinguish the James-Schreier spaces from the James spaces? Obviously, if they were isomorphic, our findings would be of no interest. In order to resolve this problem, we turned to Banach-space properties and, as we shall see, at that level differences abound; this is the main theme of the present paper.

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A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

Koszmider

Laustsen

2021

Advances in Mathematics

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We consider the closed subspace of ∞ generated by c 0 and the characteristic functions of elements of an uncountable, almost disjoint family A of infinite subsets of N. This Banach space has the form C 0 (K A ) for a locally compact Hausdorff space K A that is known under many names, including Ψ-space and Isbell-Mrówka space.We construct an uncountable, almost disjoint family A such that the algebra of all bounded linear operators on C 0 (K A ) is as small as possible in the precise sense that every bounded linear operator on C 0 (K A ) is the sum of a scalar multiple of the identity and an operator that factors through c 0 (which in this case is equivalent to having separable range). This implies that C 0 (K A ) has the fewest possible decompositions: whenever C 0 (K A ) is written as the direct sum of two infinite-dimensional Banach spaces X and Y, either X is isomorphic to C 0 (K A ) and Y to c 0 , or vice versa. These results improve previous work of the first named author in which an extra set-theoretic hypothesis was required. We also discuss the consequences of these results for the algebra of all bounded linear operators on our Banach space C 0 (K A ) concerning the lattice of closed ideals, characters and automatic continuity of homomorphisms.To exploit the perfect set property for Borel sets as in the classical construction of an almost disjoint family by Mrówka, we need to deal with N × N matrices rather than with the usual partitioners of an almost disjoint family. This noncommutative setting requires new ideas inspired by the theory of compact and weakly compact operators and the use of an extraction principle due to van Engelen, Kunen and Miller concerning Borel subsets of the square.

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Maximal Ideals in the Algebra of Operators on Certain Banach Spaces

Cited by 26 publications

References 30 publications

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

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

An amalgamation of the Banach spaces associated with James and Schreier, Part I: Banach-space structure

A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

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