The cofinal property of the reflexive indecomposable Banach spaces

Argyros, Spiros A.; Raikoftsalis, Theocharis

doi:10.5802/aif.2697

Cited by 5 publications

(2 citation statements)

References 20 publications

(38 reference statements)

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“…This operator cannot be extended onto the whole space AH. [8] constructed another separable L ∞ counterexample to the scalar-plus-compact problem. However it contains ℓ 1 and Lemma 4.3.…”

Section: Counterexamplesmentioning

confidence: 99%

On Uniformly Finitely Extensible Banach spaces

Castillo

Ferenczi

Moreno

2014

Journal of Mathematical Analysis and Applications

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We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in Moreno-Plichko, On automorphic Banach spaces, Israel J. Math. 169 (2009) 29-45 and Castillo-Plichko, Banach spaces in various positions. J. Funct. Anal. 259 (2010) 2098-2138. We show that they have the Uniform Approximation Property of Pe lczyński and Rosenthal and are compactly extensible. We will also consider their connection with the automorphic space problem of Lindenstrauss and Rosenthal -do there exist automorphic spaces other than c 0 (I) and ℓ 2 (I)?-showing that a space all whose subspaces are UFO must be automorphic when it is Hereditarily Indecomposable (HI), and a Hilbert space when it is either locally minimal or isomorphic to its square. We will finally show that most HI -among them, the super-reflexive HI space constructed by Ferenczi-and asymptotically ℓ 2 spaces in the literature cannot be automorphic.Automorphic space problem: Does there exist an automorphic space different from c 0 (I) or ℓ 2 (I)?The papers [11, 41] and [18] considered different aspects of the automorphic space problem. In particular, the following two groups of notions were isolated: Definition 1. A couple (Y, X) of Banach spaces is said to be (compactly) extensible if for every subspace E ⊂ Y every (compact) operator τ : E → X can be extended 46B03, 46B07, 46B08, 46M18, 46B25, 46B42.

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Section: Counterexamplesmentioning

confidence: 99%

On Uniformly Finitely Extensible Banach spaces

Castillo

Ferenczi

Moreno

2014

Journal of Mathematical Analysis and Applications

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“…Our theorem can also be compared to [3] and [9], where it was shown that every separable reflexive Banach space (e.g., ℓ 2 ) is a quotient of a hereditarily indecomposable space, a highly non-classical class of Banach spaces recalled in Section 2. Analogously, by our result, many simplistic classical Banach algebras appear as quotient algebras of L(X) spaces, a class of indirectly defined, and thus more intractable, objects.…”

Section: Introductionmentioning

confidence: 99%

A hierarchy of Banach spaces with C(K) Calkin algebras

Motakis¹,

Puglisi²,

Zisimopoulou³

2016

Indiana Univ. Math. J.

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Abstract. For every well founded tree T having a unique root such that every non-maximal node of it has countable infinitely many immediate successors, we construct a L∞-space XT . We prove that for each such tree T , the Calkin algebra of XT is homomorphic to C(T ), the algebra of continuous functions defined on T , equipped with the usual topology. We use this fact to conclude that for every countable compact metric space K there exists a L∞-space whose Calkin algebra is isomorphic, as a Banach algebra, to C(K).

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Interpolator Symmetries and New Kalton-Peck Spaces

Castillo,

Corrêa,

Ferenczi

et al. 2024

Results Math

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We study the six diagrams generated by the first three Schechter interpolators $$\Delta _2(f)= f''(1/2)/2!, \Delta _1(f)= f'(1/2), \Delta _0(f)=f(1/2)$$ Δ 2 ( f ) = f ′ ′ ( 1 / 2 ) / 2 ! , Δ 1 ( f ) = f ′ ( 1 / 2 ) , Δ 0 ( f ) = f ( 1 / 2 ) acting on the Calderón space associated to the pair $$(\ell _\infty , \ell _1)$$ ( ℓ ∞ , ℓ 1 ) . We will study the remarkable and somehow unexpected properties of all the spaces appearing in those diagrams: two new spaces (and their duals), two Orlicz spaces (and their duals) in addition to the third order Rochberg space, the standard Kalton-Peck space $$Z_2$$ Z 2 and, of course, the Hilbert space $$\ell _2$$ ℓ 2 . We will also deal with a nice test case: that of weighted $$\ell _2$$ ℓ 2 spaces, in which case all involved spaces are Hilbert spaces.

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The cofinal property of the reflexive indecomposable Banach spaces

Cited by 5 publications

References 20 publications

On Uniformly Finitely Extensible Banach spaces

On Uniformly Finitely Extensible Banach spaces

A hierarchy of Banach spaces with C(K) Calkin algebras

Interpolator Symmetries and New Kalton-Peck Spaces

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