Function spaces not containing ℓ1

Argyros, Spiros A.; Manoussakis, Antonis; Petrakis, Minos

doi:10.1007/bf02776049

Cited by 5 publications

(11 citation statements)

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“…In our previous work (cf. [1]), but also in [3], we have started studying the structure of the subspaces of V 0 2 . ¿From our point of view a sufficient understanding of the structure of V 0 2 requires answers to the following problems.…”

Section: Introductionmentioning

confidence: 99%

“…Our next two problems concern non-reflexive subspaces of V 0 2 . Let us begin with a result from [3] which asserts that every non reflexive subspace X of V 0 2 contains an isomorph of ℓ 2 or c 0 . To see this we start with some f ∈ X * * \ X, where X * * is considered as a subspace of V 2 .…”

Section: Introductionmentioning

confidence: 99%

“…The case of a continuous f is more interesting. As it is shown in [3], such an f is a difference of bounded semicontinuous functions (DBSC) when f is considered as a function with domain (B (V 0 2 ) * , w * ). A result of Haydon, Odell and Rosenthal (cf.…”

Section: Introductionmentioning

confidence: 99%

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Hausdorff measures and functions of bounded quadratic variation

Apatsidis

Argyros

Kanellopoulos

2011

Trans. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hausdorff measures and functions of bounded quadratic variation

Apatsidis

Argyros

Kanellopoulos

2011

Trans. Amer. Math. Soc.

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“…Moreover every reflexive saturated subspace of V 0 2 has separable dual. As is shown in [4] every non-reflexive subspace of V 0 2 contains either c 0 or l 2 . It is not known if every reflexive subspace of V 0 2 contains some l p for 2 p < ∞.…”

Section: Theorem 3 Let T ∈ B(jt) With T * [Jt * ] Non-separable Thementioning

confidence: 92%

“…By Theorem 3.2 of [4] there exists a bounded onto linear map Q : (V 0 2 ) * → l 2 ([0, 1]) with Ker Q separable. So, since T * has non-separable range we obtain that the same holds for Q • T * : (V 0 2 ) * → l 2 ([0, 1]).…”

Section: Lemma 41 the Setmentioning

confidence: 97%

On the subspaces of JF and JT with non-separable dual

Apatsidis

Argyros

Kanellopoulos

2008

Journal of Functional Analysis

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The connectivity of the hydrophobic medium in the nonionic binary system C 12 EO 6 /H 2 O is studied by monitoring the diffusion constants of tracer molecules at the transition between the hexagonal mesophase and the fluid isotropic phase. The increase in the transverse diffusion coefficient on approaching the isotropic phase reveals the proliferation of bridge-like defects connecting the surfactant cylinders. This suggests that the isotropic phase has a highly connected structure. Indeed, we find similar diffusion coefficients in the isotropic and cubic bicontinuous phases. The temperature dependence of the lattice parameter in the hexagonal phase confirms the change in connectivity close to the hexagonal-isotropic transition. Finally, an X-ray investigation of the isotropic phase shows that its structure is locally similar to that of the hexagonal phase.

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Algebras of Diagonal Operators of the Form Scalar-Plus-Compact Are Calkin Algebras

Motakis¹,

Puglisi²,

Tolias³

2020

Michigan Math. J.

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For every Banach space X with a Schauder basis consider the Banach algebra RI ⊕ K diag (X) of all diagonal operators that are of the form λI + K. We prove that RI ⊕ K diag (X) is a Calkin algbra i.e., there exists a Banach space Y X so that the Calkin algebra of Y X is isomorphic as a Banach algebra to RI ⊕ K diag (X). Among other applications of this theorem we obtain that certain hereditarily indecomposable spaces and the James spaces Jp and their duals endowed with natural multiplications are Calkin algebras, that all non-reflexive Banach spaces with unconditional bases are isomorphic as Banach spaces to Calkin algebras, and that sums of reflexive spaces with unconditional bases with certain James-Tsirelson type spaces are isomorphic as Banach spaces to Calkin algebras.where the supremum is taken over all disjoint collections of intervals of N, is finite.The spaces J p are quasi-reflexive of order one. It was for observed by A. D. Andrew and W. L. Green in [AG] that J p , after appropriate renorming, becomes a non-unital Banach algebra when endowed with coordinate-wise multiplication with respect to the basis e 1 , e 2 − e 1 , e 3 − e 2 , . . .. We denote the unitization of James space by Re ω ⊕ J p , for 1 < p < ∞.Theorem III. The spaces Re ω ⊕ J p , 1 < p < ∞ are Calkin algebras.S. F. Bellenot, R. G. Haydon, and E. Odell in [BHO] extended the definition of James, based on the unit vector basis of ℓ p , to an arbitrary space X with an unconditional basis to define the space J(X), the "jamesification" of X. The space J(X) is quasi-reflexive of order one whenever X is reflexive. As it so happens this space is a non-unital Banach algebra as well the unitization of which we denote by Re ω ⊕ J(X). Furthermore, a special subspace of J * (X) which we denote by J * (X), and coincides with J(X) * when X does not contain ℓ 1 , is a separable unital Banach algebra.Theorem IV. For any space X with a normalized unconditional basis the spaces Re ω ⊕ J(X) and J * (X) are Calkin algebras.Next we turn our attention to spaces with unconditional bases endowed with coordinate-wise multiplication. We study these spaces themselves as Banach algebras. We do not prove that they are Calkin algebras but they always embed in them as complemented ideals. In what follows "⊕" denotes the direct sum of Banach spaces.Theorem V. Let A be Banach space with a normalized unconditional basis endowed with coordinate-wise multiplication and B be a Banach algebra. In the cases described in the list bellow there exists a Banach space Y so that the Calkin algebra Cal(Y) contains an idealÃ isomorphic as a Banach algebra to A and a subalgebrã B isomorphic as a Banach algebra to B so that Cal(Y) =Ã ⊕B.(a) The space B is C(ω), the space of convergent scalar sequences with pointwise multiplication.

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Function spaces not containing ℓ1

Cited by 5 publications

References 15 publications

Hausdorff measures and functions of bounded quadratic variation

Hausdorff measures and functions of bounded quadratic variation

On the subspaces of JF and JT with non-separable dual

Algebras of Diagonal Operators of the Form Scalar-Plus-Compact Are Calkin Algebras

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