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~<~

, if the above approach proves that A is primary, then

the technique of [10] should show that H ~ is primary. As a word of warning concerning the naivet& of the conjecture, let us mention that the only complemented subspaces of A which are known are either isomorphic to A, A e y, or to X e Y, where X is isomorphic to a complemented subspace of C[0,

I] and

Y~@[~0

with d i m E n < ~ for all n = I, 2,

Much less seems to be known about the subspaces of H I. It is also not known if H ~ is primary. To prove that H I is primary, first consider the question: Is H l isomorphic to + p. G. CASAZZA.

2098

Department of Mathematics,

The University of Alabama in Huntsville.

C~

H I" )~i ?

Next, look at Billard's basis for H l [11].

Since this basis is even more di-

rectly related to the Haar system than the basis for A, this question could actually prove to be easier than the others (again using the techniques of [8, 10]). ~Editor's Note. and ( ~

Hr

i

After this paper had been submitted Wojtaszczyk proved that (~_oA)c~A Thus the first step in the program proposed here to prove that A and H I

are primary has already been made. Wojtaszcyk also answered some other old questions concerning the spaces A and H I. For example, he proved that the linear group of the space A is contractible. His method seems to work for the spaces

A ( ~ n)

and H~I~ n) as well. LITERATURE CITED

I. 2. 3. 4. 5. 6. 7. 8. 9. 10. !I.

P. Enflo, "A counterexample to the approximation property in Banach spaces," Acta. Math., 130, 309-317 (1973). A. M. Davie, "The approximation problem for Banach spaces," Bull. London Math. Soc., No. 5, 261-266 (1973). T. Figiel, "Further counterexamples to the approximation problem," dittoed notes. A. Pelczy~ski, "Banach spaces of analytic functions and absolutely surmning operators," CBMS Regional Conf. Ser. Math., No. 30 (1977). J. Lindenstrauss and A. Pelczy~ski, "Contributions to the theory of classical Banach spaces," J. Funct. Anal., No. 8, 225-249 (1971). S. V. Bochkarev, "Existence of a basis in the space of functions analytic in the disk and some properties of Franklin's system," Mat. Sb., 95, No. I, 3-18 (1974). B. S. Mityagin, "The homotopy structure of a linear group of a Banach space," Usp. Mat. Nauk, 25, No. 5, 63-106 (1970). D. Alspach, P. Enflo, and E. Odell, "On the structure of separable ~p spaces (1 < p < ~)," Stud. Math., 60, 79-90 (1977). P. Wojtaszczyk, "On projections in spaces of bounded analytic functions with applications," Preprint 115, Inst. Math., Polish Academy of Sciences. P. G. Casazza, C. Kottman, and B. L. Lit, "On some classes of primary Banach spaces," (preprint). P. Billard, "Bases dana H I e t bases de sous espaces de dimension finie dana A," Proc. Conf. Oberwolfach, August 14-22, 1971, ISNM Vol. 20, Birkh~user Verlag, Basel--Stuttgart (1972).

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