A construction of a Banach space C(K) with few operators

Plebanek, Grzegorz

doi:10.1016/j.topol.2004.03.001

Cited by 44 publications

(45 citation statements)

References 20 publications

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“…On the other hand, there exist Banach spaces in which only finitedimensional and co-finite-dimensional subspaces are complemented (Gowers and Maurey [11]). There even exist (necessarily, non-separable) C(K) spaces with no non-trivial bounded projections (Koszmider [21] and Plebanek [33]). In the positive direction, one has to mention the work of Heinrich and Mankiewicz [15] where, using substructures of ultrapowers of Banach spaces, the authors show in particular the existence of non-trivial bounded projections in every dual Banach space of density greater than the continuum (see [38] for an elementary proof).…”

Section: Introductionmentioning

confidence: 99%

Banach spaces with projectional skeletons

Kubiś¹

2009

Journal of Mathematical Analysis and Applications

145

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A projectional skeleton in a Banach space is a sigma-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Sigma-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of known results concerning weakly compactly generated spaces and Plichko spaces. We prove a preservation result for Plichko Banach spaces, involving transfinite sequences of projections. As a corollary, we show that a Banach space is Plichko if and only if it has a commutative projectional skeleton.Comment: 32 pages (including index and toc); revised (added example, comments, references

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Section: Introductionmentioning

confidence: 99%

Banach spaces with projectional skeletons

Kubiś¹

2009

Journal of Mathematical Analysis and Applications

145

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“…Theorem 2.14. (Koszmider [17], Plebanek [20]) There exists an infinite compact connected Hausdorff space K 0 such that every T ∈ L(C(K 0 )) can be written as T = gI + S, where g ∈ C(K 0 ) and S is a weakly compact operator on C(K 0 ). Moreover, the space C(K 0 ) is indecomposable.…”

Section: Theorem 213 Suppose That the Space X Has The Dpp And The Rmentioning

confidence: 99%

Intrinsic characterizations of perturbation classes on some Banach spaces

Aiena

2010

Arch. Math.

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“…We now use some spaces constructed by Plebanek [13] to give C(K) examples of even and odd Banach spaces. Similar C(K) spaces were first constructed by P. Koszmider [11] under the Continuum hypothesis.…”

Section: Proposition 16 the Direct Sum Of Two Infinite-dimensional Ementioning

confidence: 99%

“…We prove that there exist even infinite-dimensional Banach spaces, using various examples from [5], including an HI and an unconditional example, Theorem 14. Moreover we use spaces constructed in [13] to give examples of even and odd spaces of the form C(K), Theorem 18. We also show that the direct sum of essentially incomparable infinite-dimensional spaces is even whenever both spaces are even, Proposition 16.…”

Section: Introductionmentioning

confidence: 99%

Even infinite-dimensional real Banach spaces

Ferenczi

Galego

2007

Journal of Functional Analysis

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This article is a continuation of a paper of the first author [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462-488] about complex structures on real Banach spaces. We define a notion of even infinite-dimensional real Banach space, and prove that there exist even spaces, including HI or unconditional examples from [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462-488] and C(K) examples due to Plebanek [G. Plebanek, A construction of a Banach space C(K) with few operators, Topology Appl. 143 (2004) 217-239]. We extend results of [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462-488] relating the set of complex structures up to isomorphism on a real space to a group associated to inessential operators on that space, and give characterizations of even spaces in terms of this group. We also generalize results of [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462-488] about totally incomparable complex structures to essentially incomparable complex structures, while showing that the complex version of a space defined by S. Argyros and A. Manoussakis [S. Argyros, A. Manoussakis, An indecomposable and unconditionally saturated Banach space, Studia Math. 159 (1) (2003) 1-32] provides examples of essentially incomparable complex structures which are not totally incomparable.

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A construction of a Banach space C(K) with few operators

Cited by 44 publications

References 20 publications

Banach spaces with projectional skeletons

Banach spaces with projectional skeletons

Intrinsic characterizations of perturbation classes on some Banach spaces

Even infinite-dimensional real Banach spaces

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