Embeddings of operator ideals into Lp-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 Lp(H), 1≤p<2. We prove that I isomorphically embeds into an Lp-space Lp(R), 1≤p<2 (here, R is the hyperfinite II1-factor) if its commutative core (that is, Calkin space for I) isomorphically embeds into Lp(0,1). Furthermore, we prove that an Orlicz ideal LM(H)≠Lp(H) isomorphically embeds into Lp(R), 1≤p<2, if and only if it is an interpolation space for the Banach couple (Lp(H),L2(H)). Finally, we consider isomorphic embeddings of (I,‖⋅‖I) into Lp-spaces associated with arbitrary finite von Neumann algebras.