Complemented subspaces of A, H1, and H

4.1.

COMPLEMENTED

SUBSPACES OF A, H ~, AND H ~+

Enflo's counterexample to the approximation problem [I], and subsequent results by Davie [2] and Figiel [3], indicate that an isomorphic classification of all closed subspaces of a Banach space X (X not isomorphic to a Hilbert space) is probably impossible in the near future. An important and difficult, but not impossible, problem is the classification of the complemented subspaces of X. Because of the recent advances in the study of the Banach space properties of the disc algebra A, H l, and H ~ (see [4]), I think we can now give serious consideration to classifying their complemented subspaces. As a first step in the process, I make the following conjecture: Conjecture:

A and H ~ are primary:

A Banach space X is primary if whenever X % Y 9 Z then either X ~ Y or X % Z. In support of the conjecture we Jill prove that if A % Y e Z and if Y is isomorphic to a complemented subspace of C[0, I] then A ~ Z. We first use an observation of Kisljakov which states that if A % Y 9 Z and if Y is isomorphic to a complemented subspace of C[0, I] then Z* is nonseparable. To see this, we let P be a projection of A onto Y and use an argument similar to the proof of Corollary 8.5 (e) of [4] to show that P*ILI/H~ maps weakly Cauchy sequences to norm convergent sequences. If Z* is separable, it is known that weak and norm convergent sequences in Z* coincide, and hence it follows that the same is true in LI/H~, which is a contradiction. It now follows by Corollary 8.5 (b) of [4] that C[0, I] is isomorphic to a complemented subspace of Z, i.e., Z % C[O, I] 9 W for some space W. Since C[0, I] is primary [5], it follows that Y 9 C[0, I] % C[0, I]. Hence A ~ Y e Z % Y 9 C[0, I] 9 W % C[0, I] W%Y. Bochkarev

[6] has shown that A has a basis consisting o f the Franklin system in ~ R [ ~ ] .

(Here we are identifying A with the subspace of C[--~, ~] spanned by the characters {einX}n~0.) If we let H n be the span of the first n elements of this basis, Delbaen has recently announced that C ~ H ~ ) C ,

is isomorphic

to a complemented

subspace of A. This subspace is par-

ticularly interesting because it is not isomorphic to A, and it is also not isomorphic to a complemented subspace of C[0, I]. The complement of this subspace is unknown and identifying it should be the first step in proving (or disproving) the conjecture. We now outline one approach which might be used to try to prove the conjecture. If X X 9 X, then X is primary if and only if X satisfies: (I) If X ~ Y 9 Z, then either Y or Z has a complemented subspace isomorphic to X; and (2) if Y is a Banach space and if X and Y are isomorphic to complemented spaces of each other, then X % Y. By Pelczynski's decomposition method,

if X ~ X ) s

, where E is co or Ip, I ~ p ~ ~, then property

Therefore, you should first consider the question of Mityagin To give a positive answer to this question,

(I) implies property

(2).

[7]: Is A isomorphic to ( ~ . ~ ) % ?

it suffices to show that

~|

is isomorphic

to a complemented subspace of A. In this case then, you need to carefully examine the construction of Delbaen. Next, you should try to generalize the technique of [8] to the basis of A, or produce a new basis of A for which the technique works. This approach to the problem has the advantage that it may immediately imply that H ~ is primary. Since Wojtaszczyk [9] has shown that ~ C ~ . ~ ) 6

~ ~C~