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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 $ X$ embeds in this weaker sense into Pisier's operator space $ \mathrm {OH}$, then $ X$ 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}} $$ X $ $ \ mathrm {OH} $$ X $ $ \ mathrm {OH} $
更新日期:2021-02-08
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