Completely coarse maps are ℝ-linear,Proceedings of the American Mathematical Society

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Completely coarse maps are ℝ-linear
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2021-01-21 , DOI: 10.1090/proc/15289
Bruno M. Braga , Javier Alejandro Chávez-Domínguez

Abstract:A map between operator spaces is called completely coarse if the sequence of its amplifications is equi-coarse. We prove that all completely coarse maps must be ${\mathbb{R}}$ -linear. On the opposite direction of this result, we introduce a notion of embeddability between operator spaces and show that this notion is strictly weaker than complete ${\mathbb{R}}$ -isomorphic embeddability (in particular, weaker than complete ${\mathbb{C}}$ -isomorphic embeddability). Although weaker, this notion is strong enough for some applications. For instance, we show that if an infinite dimensional operator space

embeds in this weaker sense into Pisier's operator space $\mathrm {OH}$ , then

must be completely isomorphic to $\mathrm {OH}$ .

中文翻译：

完全粗糙的图是ℝ-线性的

摘要：如果算子空间之间的映射的放大序列是等粗的，则称其为完全粗糙。我们证明所有完全粗糙的图都必须是线性的。在与该结果相反的方向上，我们引入了算子空间之间的可嵌入性的概念，并表明该概念比完全同构的可嵌入性（特别是弱于完全同构的可嵌入性）弱。尽管较弱，但对于某些应用程序，此概念足够强大。例如，我们表明，如果在这种较弱的意义上将无穷维算子空间嵌入Pisier的算子空间，则它必须与完全同构。 ${\ mathbb {R}}$ ${\ mathbb {R}}$ ${\ mathbb {C}}$

$\ mathrm {OH}$

更新日期：2021-02-08

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