Unconditional basic sequences and homogeneous hilbertian subspaces of non-commutative L_p spaces

Suppose A is a von Neumann algebra with a normal faithful normalized trace T. We prove that if Eis a homogeneous Hilbertian subspace of L_P(T) (1 ≤ p < ∞) such that the norms induced on E by L_p (τ) and L₂(τ) are equivalent, then E is completely isomorphic to the subspace of L_p([0, 1]) spanned by Rademacher functions. Consequently, any homogeneous subspace of L_p(τ) is completely isomorphic to the span of Rademacher functions in L_p([0, 1]). In particular, this applies to the linear span of operators satisfying the canonical anti-commutation relations. We also show that the real interpolation space (R, C)_θ,p embeds completely isomorphically into L_p(R.) (R is the hyperfinite II₁ factor) for any 1 ≤ p < 2 and θ ε (0, 1). Indiana University Mathematics Journal