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Generalized-lush spaces and the Mazur-Ulam property

Dongni Tan, Xujian Huang, Rui Liu (2013)

Studia Mathematica

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...

Geometric characterization of L₁-spaces

Normuxammad Yadgorov, Mukhtar Ibragimov, Karimbergen Kudaybergenov (2013)

Studia Mathematica

The paper is devoted to a description of all real strongly facially symmetric spaces which are isometrically isomorphic to L₁-spaces. We prove that if Z is a real neutral strongly facially symmetric space such that every maximal geometric tripotent from the dual space of Z is unitary, then the space Z is isometrically isomorphic to the space L₁(Ω,Σ,μ), where (Ω,Σ,μ) is an appropriate measure space having the direct sum property.

Geometric, spectral and asymptotic properties of averaged products of projections in Banach spaces

Catalin Badea, Yuri I. Lyubich (2010)

Studia Mathematica

According to the von Neumann-Halperin and Lapidus theorems, in a Hilbert space the iterates of products or, respectively, of convex combinations of orthoprojections are strongly convergent. We extend these results to the iterates of convex combinations of products of some projections in a complex Banach space. The latter is assumed uniformly convex or uniformly smooth for the orthoprojections, or reflexive for more special projections, in particular, for the hermitian ones. In all cases the proof...

Geometry of Banach spaces and biorthogonal systems

S. Dilworth, Maria Girardi, W. Johnson (2000)

Studia Mathematica

A separable Banach space X contains 1 isomorphically if and only if X has a bounded fundamental total w c 0 * -stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total w c 0 * -biorthogonal system.

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