Question

There are 'canonical' examples of maps on operator spaces which are not completely bounded. Nevertheless, I couldn't produce any examples of bounded projections on relatively easy to understand operator spaces which are not c.b. In particular, I failed this task for $\mathcal{K}(H)$ and $\mathcal{B}(H)$, where $H$ is a Hilbert space (I tried projections onto certain subspaces isomorphic to $c_0$ and $\ell_\infty$, respectively).

I would appreciate any examples.

Answer 1

Symmetrisation (or anti-symmetrisation).

That is: let $T:K(H)\to K(H)$ be the transpose map and let $P=({\rm id}+T)/2$. Then $P:K(H)\to K(H)$ is a projection onto the subspace of symmetric [NOT self-adjoint] compact operators. Since $T$ is not completely bounded, $P$ is not completely bounded.

This argument also shows that the symmetrisation map $M_n\to M_n$ is a norm-one projection which has cb norm $\geq (n-1)/2$.

The examples you tried fail to work because of a general result: any bounded linear map into a minimal operator space will be automatically completely bounded. This is somewhere in the first third of Effros-Ruan's book, for example, though I don't have my copy here.