Embeddings of symmetric operator spaces into L_p-spaces, 1 ≤ p < 2, on finite von Neumann algebras

Let E(0, ∞) be a symmetric function space on (0, ∞) such that the set E(0, ∞) ∩L_∞(0, ∞) is distinct from the set L_p(0, ∞)∩L_∞(0, ∞), 1 ≤ p < 2, and let E(M) be the corresponding symmetric operator space associated with an atomless semifinite σ-finite von Neumann algebra M equipped with a semifinite infinite faithful normal trace τ. We show that there exists a noncommutative probability space (N,σ) such that E(0, ∞) embeds into Lp(N) if and only if there exists a noncommutative probability space (N^,σ^) such that E(M) embeds into Lp(N^). We also establish a discrete version of this result for symmetric sequence space ℓ_E. These extend and complement earlier results in [37,40,41,55].