Coarse geometry of operator spaces and complete isomorphic embeddings into ℓ1 and c0-sums of operator spaces

The nonlinear geometry of operator spaces has recently started to be investigated. Many notions of nonlinear embeddability have been introduced so far, but, as noticed before by other authors, it was not clear whether they could be considered “correct notions”. The main goal of these notes is to provide the missing evidence to support that almost complete coarse embeddability is “a correct notion”. This is done by proving results about the complete isomorphic theory of ℓ 1 -sums of certain operators spaces. Several results on the complete isomorphic theory of c 0 -sums of operator spaces are also obtained.