  We construct nontrivial examples of weak-$C_p$ ($1\leq p \leq \infty$)
operator spaces with the local operator space structure very close to $C_p =
[R, C]_{\frac{1}{p}}$. These examples are non-homogeneous Hilbertian operator
spaces, and their constructions are similar to that of 2-convexified
Tsirelson's space by W. B. Johnson.
