m at h . FA ] 1 5 M ay 2 00 0 BANACH EMBEDDING PROPERTIES OF NON - COMMUTATIVE L p - SPACES

Let N and M be von Neumann algebras. It is proved that L p (N) does not Banach embed in L p (M) for N infinite, M finite, 1 ≤ p < 2. The following considerably stronger result is obtained (which implies this, since the Schatten p-class Cp embeds in L p (N) for N infinite). Theorem. Let 1 ≤ p < 2 and let X be a Banach space with a spanning set (x ij) so that for some C ≥ 1, (i) any row or column is C-equivalent to the usual ℓ 2-basis, (ii) (x i k ,j k) is C-equivalent to the usual ℓ p-basis, for any i 1 < i 2 < · · · and j 1 < j 2 < · · ·. Then X is not isomorphic to a subspace of L p (M), for M finite. Complements on the Banach space structure of non-commutative L p-spaces are obtained, such as the p-Banach-Saks property and characterizations of subspaces of L p (M) containing ℓ p isomorphically. The spaces L p (N) are classified up to Banach isomorphism, for N infinite-dimensional, hyperfinite and semifinite, 1 ≤ p < ∞, p = 2. It is proved that there are exactly thirteen isomorphism types; the corresponding embedding properties are determined for p < 2 via an eight level Hasse diagram. It is also proved for all 1 ≤ p < ∞ that L p (N) is completely isomorphic to L p (M) if N and M are the algebras associated to free groups, or if N and M are injective factors of type III λ and III λ ′ for 0 < λ, λ ′ ≤ 1.