Completely coarse maps are 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 R {\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 R {\mathbb {R}} -isomorphic embeddability (in particular, weaker than complete C {\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 X embeds in this weaker sense into Pisier’s operator space O H \mathrm {OH} , then X X must be completely isomorphic to O H \mathrm {OH} .