Abstract: It is known that, on the set of subspaces of a given separable Banach space (equipped with its Effros-Borel structure), the relation of isomorphism is analytic. Moreover, on the set of subspaces of $C[0,1]$, this relation is complete analytic - in a sense, $C[0,1]$ is the "most complicated" separable Banach space. On the opposite end of the scale, the Hilbert space is the "simplest" one - all of its infinite dimensional subspaces are isomorphic to each other. For other Banach spaces, very little is known about the complexity of the isomorphism relation between subspaces. In this joint work with C.Rosendal, we asked the same question about the relation of complete isomorphism between operator spaces. We constructed a separable operator space where this relation is $K_\sigma$. Do "simpler" operator spaces exist? A possible candidate will be mentioned. The talk does not require extensive knowledge of operator spaces. This if a joint work with C. Rosendal.