Subprojective Banach spaces

Oikhberg, Timur; Spinu, Eugeniu

doi:10.1016/j.jmaa.2014.11.008

Cited by 20 publications

(19 citation statements)

References 23 publications

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“…A Banach space

X

is said to be complementably homogeneous whenever for every closed subspace

Y

X

such that

Y \approx X

, there exists another closed subspace

Z

such that

Z

is complemented in

X

Z \approx X

, and

Z \subseteq Y

. A Banach space

X

is said to be uniformly subprojective (see ) if there is a constant

C

such that for every infinite‐dimensional subspace

Y \subseteq X

there is a further subspace

Z \subseteq Y

that is

C

‐complemented in

X

.…”

Section: Introduction Notation and Terminologymentioning

confidence: 99%

“…For instance, if

X

and

Y

are Banach spaces,

Y

is subprojective, and

T \in L (X, Y)

, then the dual operator

T^{*}

is strictly singular only if

T

is. If furthermore

X = Y

and is reflexive, the converse holds, that is, if

X

is reflexive then

T \in L (X)

is strictly singular if and only if

T^{*}

is (see for further discussion of subprojectivity). Complementable homogeneity, meanwhile, has been used in several recent papers (cf., for example, ) to show the existence of a unique maximal ideal in

L (X)

.…”

Section: Introduction Notation and Terminologymentioning

confidence: 99%

See 1 more Smart Citation

Garling sequence spaces

Albiac

Ansorena

Wallis

2018

J. London Math. Soc.

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The aim of this paper is to introduce and investigate a new class of separable Banach spaces modeled after an example of Garling from 1968. For each 1⩽p<∞ and each nonincreasing weight w∈c0∖ℓ1, we exhibit an ℓp‐saturated, complementably homogeneous, and uniformly subprojective Banach space g(w,p). We also show that g(w,p) admits a unique subsymmetric basis despite the fact that for a wide class of weights it does not admit a symmetric basis. This provides the first known examples of Banach spaces where those two properties coexist.

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“…A Banach space

X

is said to be complementably homogeneous whenever for every closed subspace

Y

X

such that

Y \approx X

, there exists another closed subspace

Z

such that

Z

is complemented in

X

Z \approx X

, and

Z \subseteq Y

. A Banach space

X

is said to be uniformly subprojective (see ) if there is a constant

C

such that for every infinite‐dimensional subspace

Y \subseteq X

there is a further subspace

Z \subseteq Y

that is

C

‐complemented in

X

.…”

Section: Introduction Notation and Terminologymentioning

confidence: 99%

“…For instance, if

X

and

Y

are Banach spaces,

Y

is subprojective, and

T \in L (X, Y)

, then the dual operator

T^{*}

is strictly singular only if

T

is. If furthermore

X = Y

and is reflexive, the converse holds, that is, if

X

is reflexive then

T \in L (X)

is strictly singular if and only if

T^{*}

L (X)

.…”

Section: Introduction Notation and Terminologymentioning

confidence: 99%

Garling sequence spaces

Albiac

Ansorena

Wallis

2018

J. London Math. Soc.

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“…The nonisomorphism result will not be required here, and we only refer to the analogous argument for countable direct sums [23,Theorem 2] (cf. also the proof of [30,Proposition 2.2]).…”

Section: Johnson [18 Theorem 1] and Figielmentioning

confidence: 90%

The Quotient Algebra of Compact-by-Approximable Operators on Banach Spaces Failing the Approximation Property

Tylli

Wirzenius²

2019

J. Aust. Math. Soc.

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We initiate a study of structural properties of the quotient algebra K(X)/A(X) of the compact-by-approximable operators on Banach spaces X failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from c 0 into K(Z)/A(Z), where Z belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a non-separable space c 0 (Γ) into K(Z F J )/A(Z F J ), where Z F J is a universal compact factorisation space arising from the work of Johnson and Figiel.

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“…If there exists µ < λ such that the restriction P µ | M is not strictly singular, then there exists an infinitedimensional subspace N ⊆ M such that P µ | N is an isomorphism. Since the range of P µ is isometric to C([0, µ], X), which is subprojective by our induction hypothesis, N contains an infinite-dimensional subspace complemented in C 0 ([0, λ], X) [28,Corollary 2.4].…”

Section: Corollary 26 a L 1 -Space Is Subprojective If And Only If mentioning

confidence: 99%