Completely coarse maps are ℝ-linear

Braga, Bruno; Chávez-Domínguez, Javier

doi:10.1090/proc/15289

Completely coarse maps are ${\mathbb {R}}$-linear
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by Bruno M. Braga and Javier Alejandro Chávez-Domínguez PDF

Proc. Amer. Math. Soc. 149 (2021), 1139-1149 Request permission

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 $\mathrm {OH}$.

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