TITLE: What Approximation Property does the space of Schatten-p class operators have?
QUESTION [5 upvotes]: Background
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?

Thank you!
Reference
Good reference for different approximation properties: Handbook of the Geometry of Banach Spaces, vol. 1 -- contribution by Pete Casazza

REPLY [8 votes]: It has a Schauder basis, namely $e_i\otimes e_j$, where $(e_i)$ is an orthonormal basis of the Hilbert space. This holds for $1\le p<\infty$. For the Schatten ideal of compact operators, I do not know the answer.