Banach spaces having the Radon-Nikodym property and numerical index 1

Martı́n, Miguel

doi:10.1090/s0002-9939-03-07176-4

“…As we have already mentioned, at the beginning of Section 2, the main examples of Banach spaces with numerical index 1 are known to be almost-CL-spaces. Actually, it is proved in [23] that every Banach space with numerical index 1 and the Radon-Nikodým property is an almost-CL-space and it satisfies that…”

Section: Remark 3•2mentioning

confidence: 99%

Numerical index of Banach spaces and duality

Boyko

¹

,

Kadets

²

,

Martı́n

³

et al. 2007

Math. Proc. Camb. Phil. Soc.

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“…The c 0 and l 1 sums of CL-spaces are also proved to be CL-spaces (see [15]). For more information on CL-spaces, we refer to [1,6,11,14,20,22,23]. Let us mention that the key part in handling Question 1.1 is how to construct or find a proper linear isometry with the desired property.…”

Section: Introductionmentioning

confidence: 99%

RETRACTED – The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

Tan¹,

Huang²

2021

Proceedings of the Edinburgh Mathematical Society

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We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

show abstract

“…The c 0 and l 1 sums of CL-spaces are also proved to be CL-spaces (see [15]). For more information on CL-spaces, we refer to [1,6,11,14,20,22,23]. Let us mention that the key part in handling Question 1.1 is how to construct or find a proper linear isometry with the desired property.…”

Section: Introductionmentioning

confidence: 99%

The wigner property for CL-spaces and finite-dimensional polyhedral Banach spaces

Tan¹,

Huang²

2021

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

show abstract

Banach spaces having the Radon-Nikodym property and numerical index 1

Abstract: Abstract. Let X be a Banach space with the Radon-Nikodỳm property. Then, the following are equivalent.

Cited by 18 publications

References 14 publications

Numerical index of Banach spaces and duality

Numerical index of Banach spaces and duality

RETRACTED – The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

The wigner property for CL-spaces and finite-dimensional polyhedral Banach spaces

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