COℒ_p spaces - The local structure of non-commutative L_p spaces

Developing the theory of COℒ_p spaces (a variation of the non-commutative analogue of L_p spaces), we provide new tools to investigate the local structure of non-commutative L_p spaces. Under mild assumptions on the underlying von Neumann algebras, non-commutative L_p spaces with Grothendieck's approximation property behave locally like the space of matrices equipped with the p-norm (of the sequences of their singular values). As applications, we obtain a basis for non-commutative L_p spaces associated with hyperfinite von Neumann algebras with separable predual von Neumann algebras generated by free groups, and obtain a basis for separable nuclear C*-algebras.