Complemented Subspaces of Spaces Obtained by Interpolation

If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A0, A1) such that A0 and A1 are isometric to X ⊕ V , and any intermediate space obtained using the real or complex interpolation method contains a complemented subspace isomorphic to Z. Thus many properties of Banach spaces, including having non-trivial cotype, having the Radon–Nikodym property, and having the analytic unconditional martingale difference sequence property, do not pass to intermediate spaces. There are many Banach space properties that pass to spaces obtained by the complex method of interpolation. For example, it is known that if a couple (A0, A1) is such that A0 and A1 both have the UMD (unconditional martingale difference sequence) property, and if Aθ is the space obtained using the complex interpolation method with parameter θ, then Aθ has the UMD property whenever 0 < θ < 1. Another example is type of Banach spaces: if A0 has type p0 and A1 has type p1, then Aθ has type pθ, where 1/pθ = (1−θ)/p0 +θ/p1. Similar results are true for the real method of interpolation. If we denote by Aθ,p the space obtained using the real interpolation method from a couple (A0, A1) with parameters θ and p, then Aθ,p has the UMD property whenever A0 and A1 have the UMD property, 0 < θ < 1, and 1 < p <∞. Similarly, if A0 has type p0 and A1 has type p1, then Aθ,p has type pθ, where 1/pθ = (1− θ)/p0 + θ/p1 and p = pθ (see [5] 2.g.22). However, there are other properties for which it has been hitherto unknown whether they pass to the intermediate spaces. Examples include the Radon–Nikodym property, the AUMD (analytic unconditional martingale difference sequence) property, and having non-trivial cotype. The second named author was supported in part by N.S.F. Grant DMS 9001796. A.M.S. (1980) subject classification: 46B99. SPACES OBTAINED BY INTERPOLATION This paper deals with these properties, showing that they do not pass to the intermediate spaces. Indeed, we show the surprising fact that there is a couple (A0, A1) such that A0 and A1 are both isometric to l1, but all the real or complex intermediate spaces contain a complemented subspace isomorphic to c0. This improves a result due to Pisier, who gave an example of a couple (A0, A1) for which A0 is isometric to L1, A1 is isometric to a dense subspace of c0, and c0 is finitely represented in every intermediate space Aθ obtained by the complex interpolation method (see [3]). Notation Here we outline the notation we will use about interpolation couples. The reader is referred to [1] or [2] for details. A Banach couple is a pair of Banach spaces (A0, A1) such that A0 and A1 both embed into a common topological vector space, Ω, which we shall call the ambient space. Given such a couple, we define Banach spaces A0 +A1 (with norm ‖x‖ = inf{ ‖x0‖A0 + ‖x1‖A1 : x0 ∈ A0, x1 ∈ A1, x0 + x1 = x}) and A0 ∩A1 (with norm ‖x‖ = max{‖x‖A0 , ‖x‖A1}). A map between two couples T : (A0, A1)→ (B0, B1) is a linear map T : A0 +A1 → B0 +B1 such that T (A0) ⊆ B0 and T (A1) ⊆ B1, and such that ‖T‖A0→B0 , ‖T‖A1→B1 <∞. An interpolation method, I, is a functor that takes a Banach couple (A0, A1) to a single Banach space AI , such that A0 ∩A1 ⊆ AI ⊆ A0 +A1 with c−1 ‖x‖A0+A1 ≤ ‖x‖AI ≤ c ‖x‖A0∩A1 , (1) and so that if T : (A0, A1) → (B0, B1) is a map between couples, then T (AI) ⊆ BI with ‖T‖AI→BI <∞. An interpolation method is called exponential with exponent θ if 0 < θ < 1, and whenever T : (A0, A1)→ (B0, B1) is a map between couples, then ‖T‖AI→BI ≤ c ‖T‖ 1−θ A0→B0 ‖T‖ θ A1→B1 . (2) An interpolation method is called exact exponential if it is exponential and c = 1 in inequalities (1) and (2). The most well known interpolation methods are the real interpolation method, and the complex interpolation method. They are both exponential, and the complex interpolation method is exact exponential. Another interpolation method, parameterized by 0 < θ < 1, is the following: if x ∈ A0 ∩A1, then let