Completely coarse maps are $\mathbb R$-linear

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 $\mathrm{OH}$.