| Résume | A Banach space (X, ||·||) can be viewed as a metric space endowed with the metric d(x,y)=||x-y||. The nonlinear theory of Banach spaces then asks how much of the linear structure of (X,||· ||) can be recovered by looking at X simply as a metric space, i.e., leaving its linear structure aside. Although the nonlinear theory of Banach spaces has been receiving a lot of attention, its noncommutative counterpart (i.e., the nonlinear theory of operator spaces) has been being neglected until very recently. Precisely, in joint works with Alejandro Chávez-Domínguez and Thomas Sinclair, we have introduced some notions of nonlinear maps and embeddability/equivalences between operator spaces which are (1) weak enough so that their existence is strictly weaker than their linear counterparts and (2) strong enough so that they still preserve some of the linear operator structure of the spaces. In this talk, I will give an overview of the current status of the nonlinear theory of operator spaces. |
