Coarse geometry of operator spaces and complete isomorphic embeddings into $\ell_1$ and $c_0$-sums of operator spaces

TL;DR

This paper investigates the nonlinear geometry of operator spaces, demonstrating that almost complete coarse embeddability is a valid concept through results on the complete isomorphic theory of $oldsymbol{ ext{l}_1}$-sums and $oldsymbol{ ext{c}_0}$-sums of operator spaces.

Contribution

It provides the first substantial evidence supporting almost complete coarse embeddability as a correct notion in the nonlinear geometry of operator spaces.

Findings

Proves results on the complete isomorphic theory of $ ext{l}_1$-sums of operator spaces.

Obtains results on the complete isomorphic theory of $ ext{c}_0$-sums of operator spaces.

Supports the validity of almost complete coarse embeddability as a meaningful concept.

Abstract

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 \emph{almost complete coarse embeddability} is ``a correct notion''. This is done by proving results about the complete isomorphic theory of $\ell_1$-sums of certain operators spaces. Several results on the complete isomorphic theory of $c_0$-sums of operator spaces are also obtained.