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On the Lipschitz Decomposition Problem in Ordered Banach Spaces and Its Connections to Other Branches of Mathematics

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TLDR
For any Banach space X, ordered by a closed generating cone C ⊆ X, do there always exist Lipschitz functions ⋅+ : X → C and ⋆− : x → C satisfying x = x+ − x− for every x ∈ X?
Abstract
Consider the following still-open problem: for any Banach space X, ordered by a closed generating cone C ⊆ X, do there always exist Lipschitz functions ⋅+ : X → C and ⋅− : X → C satisfying x = x+ − x− for every x ∈ X?

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References
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Book

Banach Lattices and Positive Operators

TL;DR: In this paper, the authors propose the use of linear operators on positive matrices and apply it to non-positive matrices, including the case of positive projections. But they do not consider the case where positive projections are defined by a linear operator.
Book

Geometric Nonlinear Functional Analysis

TL;DR: The Radon-Nikodym property Negligible sets and Gateaux differentiability Lipschitz classification of Banach spaces Uniform embeddings into Hilbert space Uniform classification of spheres Uniform classifications of nonlinear quotient maps Oscillation of uniformly continuous functions on unit spheres of infinite-dimensional subspaces Perturbations of local and global isometries Twisted sums Group structure on Banach space as discussed by the authors.
Book

Topics in Banach space theory

TL;DR: Godefroy as discussed by the authors provides the reader with the necessary technical tools and background to reach the frontiers of research without the introduction of too many extraneous concepts, and encourages graduate students interested in learning more about Banach spaces.
Book

Introduction to Tensor Products of Banach Spaces

TL;DR: In this article, a self-contained introduction to the theory of tensor products of Banach spaces is provided for graduate students in analysis or for researchers in other fields who wish to become acquainted with this area The only prerequisite are a basic knowledge of functional analysis and measure theory.
Book

Tensor Norms and Operator Ideals

TL;DR: Tensor Norms and Operator Ideals as mentioned in this paper are a generalization of the algebraic theory of Tensor Products and have been studied extensively in the literature, e.g. in the context of algebraic theories of tensors.
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