Abstract
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.
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