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