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Generalized-lush spaces and the Mazur-Ulam property

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Tan D., Huang X., Liu R.

Studia Mathematica

2013

tom 219

nr 2

139-153

We introduce a new class of Banach spaces, called generalized-lush spaces (GL-spaces for short), which contains almost-CL-spaces, separable lush spaces (in particular, separable C-rich subspaces of C(K)), and even the two-dimensional space with hexagonal norm. We find that the space C(K,E) of vector-valued continuous functions is a GL-space whenever E is, and show that the set of GL-spaces is stable under c₀-, l₁- and $l_{∞}$-sums. As an application, we prove that the Mazur-Ulam property holds for a larger class of Banach spaces, called local-GL-spaces, including all lush spaces and GL-spaces. Furthermore, we generalize the stability properties of GL-spaces to local-GL-spaces. From this, we can obtain many examples of Banach spaces having the Mazur-Ulam property.

Upper and lower estimates for Schauder frames and atomic decompositions

100%

Beanland K., Freeman D., Liu R.

Fundamenta Mathematicae

2015

tom 231

nr 2

161-188

We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite-dimensional decompositions of Banach spaces can be extended and modified to Schauder frames. We show as well that if a separable infinite-dimensional Banach space has a Schauder frame, then it also has a Schauder frame which is not shrinking.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Studia Mathematica

2 Liu R.

1 Beanland K.

1 Freeman D.

1 Huang X.

1 Tan D.

1 2015

1 2013

Wyniki wyszukiwania

Generalized-lush spaces and the Mazur-Ulam property

Upper and lower estimates for Schauder frames and atomic decompositions