This is a follow-up question to: What (classes of) Banach spaces are known to have Schauder basis?
In the previous question, I asked about what spaces are known to have Schauder basis. It seems that not a lot of positive results are available in that area. So I am restricting my question to one of the spaces that is in particular relevant to my research:
Does $C_p$, the Banach space of all Schatten-p operators on a separable infinite-dimensional Hilbert space, have property $\pi$? Or does it have a (Schauder) basis?
Good reference for different approximation properties: Handbook of the Geometry of Banach Spaces, vol. 1 -- contribution by Pete Casazza