Towards a theory of coarse geometry of operator spaces

We introduce two notions of coarse embeddability between operator spaces: almost complete coarse embeddability of bounded subsets and spherically-complete coarse embeddability. We provide examples showing that these notions are strictly weaker than complete isomorphic embeddability—in fact, they do not even imply isomorphic embeddability. On the other hand, we show that, despite their nonlinearity, the existence of such embeddings provides restrictions on the linear operator space structures of the spaces. Examples of nonlinear equivalences between operator spaces are also provided.