A non-separable uniformly convex Banach space on which there are few operators

Wark, H. M.

doi:10.4064/sm8740-5-2017

Cited by 6 publications

(9 citation statements)

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“…where X pE W q denotes the ideal of operators on E W with separable range. Recently Wark gave a second example of such a space [6] with the additional property that the space is uniformly convex. For the rest of our paper E W can be taken to be either of these spaces.…”

Section: Resultsmentioning

confidence: 99%

“…In the notation of [5] and [6] the family tepαq : α ă ω 1 u is a transfinite basis of E W . See the proofs of Theorem 2 in [5] or Proposition 8 in [6]. It is shown in [3] that Banach spaces with transfinite bases have the approximation property.…”

Section: Resultsmentioning

confidence: 99%

“…The Banach space in question may be taken to be the dual of any reflexive, non-separable Banach space that has few operators, in the sense that every bounded operator on E is the sum of a scalar multiple of the identity and an operator with separable range. Wark has shown that such Banach spaces exist [5,6].…”

Section: Introductionmentioning

confidence: 99%

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Subspaces that can and cannot be the kernel of a bounded operator on a Banach space

Laustsen¹,

White²

2020

Banach Algebras and Applications

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Given a Banach space E, we ask which closed subspaces may be realised as the kernel of a bounded operator E Ñ E. We prove some positive results which imply in particular that when E is separable every closed subspace is a kernel. Moreover, we show that there exists a Banach space E which contains a closed subspace that cannot be realised as the kernel of any bounded operator on E. This implies that the Banach algebra BpEq of bounded operators on E fails to be weak*-topologically left Noetherian in the sense of [7]. The Banach space E that we use is the dual of one of Wark's non-separable, reflexive Banach spaces with few operators. Definition 1.1. Let A be a dual Banach algebra. We say that A is weak*-topologically left Noetherian if every weak*-closed left ideal I of A is weak*-topologically finitely-generated, i.e. there exists n P N, and there exist x 1 ,. .. , x n P I such that I " Ax 1`C x 1`¨¨¨`A x n`C x n w˚.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Subspaces that can and cannot be the kernel of a bounded operator on a Banach space

Laustsen¹,

White²

2020

Banach Algebras and Applications

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show abstract

“…In the notation of [5] and [6] the family tepαq : α ă ω 1 u is a transfinite basis of E W . See the proofs of Theorem 2 in [5] or Proposition 8 in [6].…”

Section: Resultsmentioning

confidence: 99%

Subspaces that can and cannot be the kernel of a bounded operator on a Banach space

Laustsen¹,

White²

2018

Preprint

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Given a Banach space E, we ask which closed subspaces may be realised as the kernel of a bounded operator E Ñ E. We prove some positive results which imply in particular that when E is separable every closed subspace is a kernel. Moreover, we show that there exists a Banach space E which contains a closed subspace that cannot be realised as the kernel of any bounded operator on E. This implies that the Banach algebra BpEq of bounded operators on E fails to be weak*-topologically left Noetherian in the sense of [7]. The Banach space E that we use is the dual of one of Wark's non-separable, reflexive Banach spaces with few operators.Definition 1.1. Let A be a dual Banach algebra. We say that A is weak*-topologically left Noetherian if every weak*-closed left ideal I of A is weak*-topologically finitely-generated, i.e. there exists n P N, and there exist x 1 , . . . , x n P I such that I " Ax 1 `Cx 1 `¨¨¨`Ax n `Cx n w ˚.

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“…Shelah's original example relied on an additional set-theoretic axiom, ♦, but this assumption was later removed by Shelah and Steprāns [56]. Wark [58,59] has taken this line of research further by producing a reflexive and, much more recently, a uniformly convex space with the above property. Note that the space C 0 (K A ) from [34] or Theorem 2 is another instance of a nonseparable Banach space with the property that every bounded linear operator on it is the sum of a scalar multiple of the identity and an operator with separable range.…”

Section: Consequences Of Theorem 2 Context and Open Questionsmentioning

confidence: 99%

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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A non-separable uniformly convex Banach space on which there are few operators

Cited by 6 publications

References 0 publications

Subspaces that can and cannot be the kernel of a bounded operator on a Banach space

Subspaces that can and cannot be the kernel of a bounded operator on a Banach space

Subspaces that can and cannot be the kernel of a bounded operator on a Banach space

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

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