Examples

The following classes of operator spaces are completely locally reflexive [EJR98, 6.1, 6.2], [EH85, Prop. 5.4]:

1. reflexive operator spaces (e.g. finite dimensional operator spaces $L$),
2. nuclear $C^*$- algebras (e.g. $K({\cal H})$ or commutative $C^*$- algebras),⁵⁴
3. preduals of von Neumann algebras, especially duals of $C^*$-algebras (e.g. $T({\cal H})=K({\cal H})^*=B({\cal H})_*$).

Footnotes

...⁵⁴

a survey of the theory of nuclear $C^*$-algebras is e.g. to be found in [Mur90] and [Pat88].