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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} .

In these notes, we study nonlinear embeddings between Banach spaces which are also weakly sequentially continuous. In particular, our main result implies that if a Banach space X coarsely (resp. uniformly) embeds into a Banach space Y by a weakly sequentially continuous map, then every spreading model (en)n of a normalized weakly null sequence in X satisfieswhere δ Y is the modulus of asymptotic uniform convexity of Y . Among many other results, we obtain Banach spaces X and Y so that X coarsely (resp. uniformly) embeds into Y , but so that X cannot be mapped into Y by a weakly sequentially continuous coarse (resp. uniform) embedding.