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

Abstract

Suppose A is a von Neumann algebra with a normal faithful normalized trace T. We prove that if E is a homogeneous Hilbertian subspace of Lp (τ) (1 ≤ p < ∞) such that the norms induced on E by Lp (τ) and L 2 (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 III factor) for any 1 < p < 2 and θ ∈ (0,1).