Motivated by the notion of $K$-gentle partition of unity introduced in [J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Invent. Math. 160 (2005), no. 1, 59–95] and the notion of $K$-Lipschitz retract studied in [S. I. Ohta, Extending Lipschitz and Hölder maps between metric spaces, Positivity 13 (2009), no. 2, 407–425], we study a weaker notion related to the Kantorovich–Rubinstein transport distance that we call $K$-random projection. We show that $K$-random projections can still be used to provide linear extension operators for Lipschitz maps. We also prove that the existence of these random projections is necessary and sufficient for the existence of weak${}^{*}$ continuous operators. Finally, we use this notion to characterize the metric spaces $(X,d)$ such that the free space $ℱ\left(X\right)$ has the bounded approximation propriety.

中文翻译：

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Lipschitz映射的空间与最佳传输之间的线性扩展算子

受...的观念激励 $ķ$[J. R. Lee和A. Naor，通过随机度量分区来扩展Lipschitz函数，Invent。数学。第160（2005）号。1，59–95]和 $ķ$-Lipschitz在[S. I. Ohta，在度量空间之间扩展Lipschitz和Hölder映射，Positivity 13（2009），否。[2，407–425]，我们研究了与Kantorovich-Rubinstein传输距离有关的较弱概念$ķ$-随机投影。我们证明$ķ$-随机投影仍可用于为Lipschitz映射提供线性扩展算子。我们还证明了这些随机投影的存在对于弱点的存在是必要和充分的${}^{*}$连续运营商。最后，我们使用此概念来表征度量空间$（X，d）$ 这样的自由空间 $ℱ（X）$ 具有有界的近似适当性。