Examples

$B(\H)$ is an operator space by the identification $M_n(B(\H))=B(\H^n)$. Generally, each $C^*$-algebra $A$ is an operator space if $M_n(A)$ is equipped with its unique $C^*$-norm. Closed subspaces of $C^*$-algebras are called concrete operator spaces. Each concrete operator space is an operator space. Conversely, by the theorem of Ruan , each operator space is completely isometrically isomorphic to a concrete operator space.

Commutative $C^*$-algebras are homogeneous operator spaces.

The transposition $\Phi$ on $l_2(I)$ has norm $\Vert\Phi\Vert=1$, but $\Vert\Phi\Vert _{\mathrm{cb}}=\dim l_2(I)$. If $I$ is infinite, then $\Phi$ is bounded, but not completely bounded.

If $\dim\H\geqslant 2$, then $B(\H)$ is not homogeneous [Pau86, p. 6].