Rigid Οℑ_p structures of non-commutative L _p-spaces associated with hyperfinite von Neumann algebras

This paper is devoted to the study of rigid local operator space structures on non-commutative L_p-spaces. We show that for 1 ≤ p ≠ 2 〈, a non-commutative L_p-space L_p(M) is a rigid Οℑ_p space (equivalently, a rigid Οℑ _p space) if and only if it is a matrix orderly rigid Οℑ_p space (equivalently, a matrix orderly rigid Οℑ_p, space). We also show that L_p(M) has these local properties if and only if the associated von Neumann algebra M is hyperfinite. Therefore, these local operator space properties on non-commutative L_p-spaces characterize hyperfinite von Neumann algebras.