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?
We discuss the connections of this problem to a large number of other branches of mathematics: set-valued analysis, selection theorems, the non-linear geometry of Banach spaces, Ramsey theory, Lipschitz function spaces, duality theory, and tensor products of Banach spaces. We give numerous equivalent reformulations of the problem, and, through known examples, provide circumstantial evidence that the above question could be answered in the negative.
Dedicated to the occasion of Ben de Pagter’s 65th birthday
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Notes
- 1.
For our current purpose a closed cone inside a Banach space is sufficient. See the more general definitions: [10, Definitions 2.2 and 2.3].
- 2.
Although very seldomly used in analysis, this is purely an observation about real numbers: let \(a,b\in \mathbb {R}\) be distinct real numbers and let K, ε > 0 satisfy \(0<(K-\varepsilon )<\left |a-b\right |\leq K.\) If \(c\in \mathbb {R}\) is such that \(\left |a-c\right |\leq 2^{-1}K\) and \(\left |b-c\right |\leq 2^{-1}K\) , then \(\left |c-2^{-1}(a+b)\right |\leq 2^{-1}\varepsilon \).
- 3.
Let \((x,x)\in \overline {({\mathbf {B}}_{X\oplus _{\infty }X}+C\oplus _{\infty }(-C))}^{\left \Vert \cdot \right \Vert _{\infty }}\cap \** _{\infty }\). Then there exist sequences \(((a_{n},b_{n}))\subseteq {\mathbf {B}}_{X\oplus _{\infty }X}\) and ((c n, −d n)) ⊆ C ⊕∞(−C) so that (a n, b n) + (c n, −d n) → (x, x) as n →∞. For every \(n\in \mathbb {N}\), let p n := (a n + c n) − (b n − d n) and consider the sequence \(S:=\left ((a_{n},b_{n}+p_{n})+(c_{n},-d_{n})\right )=((a_{n}+c_{n},a_{n}+c_{n}))\subseteq \** _{\infty }\). This sequence S converges to (x, x) and, since p n → 0 as n →∞, for every ε > 0 the tail of S eventually lies in \(((1+\varepsilon /\alpha ){\mathbf {B}}_{X\oplus _{\infty }X}+C\oplus _{\infty }(-C))\cap \** _{\infty }\subseteq (1+\varepsilon /\alpha )\alpha \** _{1}\), and hence \(\left \Vert (x,x)\right \Vert _{\infty }\leq (\alpha +\varepsilon ).\) But this holds for every ε > 0, so \(\left \Vert (x,x)\right \Vert _{\infty }\leq \alpha \) and therefore (x, x) ∈ α Ξ1. Because the wk-closure and \(\left \Vert \cdot \right \Vert _{\infty }\)-closure of convex sets coincide, we obtain \(\overline {({\mathbf {B}}_{X\oplus _{\infty }X}+C\oplus _{\infty }(-C))}^{\mathrm{wk}}\cap \** _{\infty }\subseteq \alpha \** _{1}.\)
- 4.
Of some relevance here is Kalton and Godefroy’s result [15, Theorem 5.3] showing that bounded approximation properties transfer between X and F(X).
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Acknowledgements
The author would like to express his thanks to the MathOverflow community, especially to Bill Johnson for bringing work described in Sect. 5 to the author’s attention.
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Messerschmidt, M. (2019). On the Lipschitz Decomposition Problem in Ordered Banach Spaces and Its Connections to Other Branches of Mathematics. In: Buskes, G., et al. Positivity and Noncommutative Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10850-2_22
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