Embeddings of operator ideals into L_p-spaces on finite von Neumann algebras

Let L(H) be the ⁎-algebra of all bounded operators on an infinite dimensional Hilbert space H and let (I,‖⋅‖_I) be an ideal in L(H) equipped with a Banach norm which is distinct from the Schatten–von Neumann ideal L_p(H), 1≤p>2. We prove that I isomorphically embeds into an L_p-space L_p(R), 1≤p>2 (here, R is the hyperfinite II₁-factor) if its commutative core (that is, Calkin space for I) isomorphically embeds into L_p(0,1). Furthermore, we prove that an Orlicz ideal L_M(H)≠L_p(H) isomorphically embeds into L_p(R), 1≤p>2, if and only if it is an interpolation space for the Banach couple (L_p(H),L₂(H)). Finally, we consider isomorphic embeddings of (I,‖⋅‖_I) into L_p-spaces associated with arbitrary finite von Neumann algebras.