C*-algebras have a quantitative version of Pełczyński's property (V)

Krulišová, Hana

doi:10.21136/cmj.2017.0242-16

Cited by 6 publications

(6 citation statements)

References 10 publications

(19 reference statements)

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“…More precisely, she proved that the space

C_{0} false(normalΩ false)

enjoys the quantitative property

{true(V_{q} true)}_{ω}^{*}

with constant π (2 in the real case) for every locally compact Hausdorff space Ω. Later, H. Krulišová generalized this result by quantifying Pfitzner's theorem. In Section 4, we introduce the concepts of quantitative Pełczyński's property

(V^{*})

of order p and quantitative Pełczyński's property ( V ) of order p .…”

Section: Introduction and Notationsmentioning

confidence: 99%

Pełczyński's property () of order p and its quantification

Chen

Chávez-Domínguez

2017

Mathematische Nachrichten

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We introduce the concepts of Pełczyński's property ( ) of order and Pełczyński's property ( * ) of order . It is proved that, for each 1 < < ∞, the James -spaces enjoys Pełczyński's property ( * ) of order and the James * -spaces * (where * denotes the conjugate number of ) enjoys Pełczyński's property ( ) of order . We prove that both 1 ( ) ( a finite positive measure) and 1 enjoy a quantitative version of Pełczyński's property ( * ). K E Y W O R D SPełczyński's property ( ) of order , Pełczyński's property ( * ) of order , quantitative Pełczyński's property ( * ) of order M S C ( 2 0 1 0 ) 46-B

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“…More precisely, she proved that the space

C_{0} false(normalΩ false)

enjoys the quantitative property

{true(V_{q} true)}_{ω}^{*}

(V^{*})

of order p and quantitative Pełczyński's property ( V ) of order p .…”

Section: Introduction and Notationsmentioning

confidence: 99%

Pełczyński's property () of order p and its quantification

Chen

Chávez-Domínguez

2017

Mathematische Nachrichten

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“…Moreover, Krulišová introduced in [101] a quantitative version of property (V ) as follows: given λ 1, a Banach space has quantitative property (V ) (with constant λ), whenever for every Banach space Y and every operator T : X → Y , one has γ(T ) λ • uc(T ), where…”

Section: Quantification Of the Grothendieck Propertymentioning

confidence: 99%

Grothendieck spaces: the landscape and perspectives

Kania

2021

Jpn. J. Math.

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In 1973, Diestel published his seminal paper Grothendieck spaces and vector measures that drew a connection between Grothendieck spaces (Banach spaces for which weak-and weak*-sequential convergences in the dual space coincide) and vector measures. This connection was developed further in his book with J. Uhl Jr. Vector measures. Additionally, Diestel's paper included a section with several open problems about the structural properties of Grothendieck spaces, and only half of them have been solved to this day.The present paper aims at synthetically presenting the state of the art at subjectively selected corners of the theory of Banach spaces with the Grothendieck property, describing the main examples of spaces with this property, recording the solutions to Diestel's problems, providing generalisations/extensions or new proofs of various results concerning Grothendieck spaces, and adding to the list further problems that we believe are of relevance and may reinvigorate a better-structured development of the theory.

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“…We refer the reader also to the subsequent paper which contains a generalization of Theorem proved hereinafter. In 1994, H. Pfitzner generalized Pełczyński's theorem – he proved that all

C^{*}

‐algebras enjoy the property (V) (see ).…”

Section: Quantitative Version Of Pełczyński's Theorem and Its Generalmentioning

confidence: 99%

“…In 1994, H. Pfitzner generalized Pełczyński's theorem – he proved that all

C^{*}

‐algebras enjoy the property (V) (see ). The paper is devoted primarily to a quantification of this result (see Theorems , 4.2 there). Theorem Let Ω be a locally compact space.…”

Section: Quantitative Version Of Pełczyński's Theorem and Its Generalmentioning

confidence: 99%

Quantification of Pełczyński's property (V)

Krulišová

2017

Mathematische Nachrichten

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A Banach space has Pełczyński's property (V) if for every Banach space every unconditionally converging operator ∶ → is weakly compact. In 1962, Aleksander Pełczyński showed that ( ) spaces for a compact Hausdorff space enjoy the property (V), and some generalizations of this theorem have been proved since then. We introduce several possibilities of quantifying the property (V). We prove some characterizations of the introduced quantitative versions of this property, which allow us to prove a quantitative version of Pelczynski's result about ( ) spaces and generalize it. Finally, we study the relationship of several properties of operators including weak compactness and unconditional convergence, and using the results obtained we establish a relation between quantitative versions of the property (V) and quantitative versions of other well known properties of Banach spaces. K E Y W O R D S Measures of weak non-compactness, Pełczyński's property (V), quantitative Pełczyński's property (V), unconditionally converging operators M S C ( 2 0 1 0 ) 46B04, 47B10, 46E15

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C*-algebras have a quantitative version of Pełczyński's property (V)

Cited by 6 publications

References 10 publications

Pełczyński's property () of order p and its quantification

Pełczyński's property () of order p and its quantification

Grothendieck spaces: the landscape and perspectives

Quantification of Pełczyński's property (V)

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