ON QUASI-COMPLEMENTED SUBSPACES OF BANACH SPACES* By HASKELL P. ROSENTHAL

DEPARTMENT OF , UNIVERSITY QF CALIFORNIA, BERKELEY Communicated by Charles B. Morrey, Jr., December 4, 1967 (1) Introduction.-Let X be a (real or complex) , and A be a closed subspace of X. A is said to be quasi-complemented if there exists a closed subspace B of X such that A nfB = {I0}, with A + B dense in X. (Such a sub- space B is called a quasi-complement for A.) From now on, the term "subspace" shall refer to a closed, infinite-dimensional subspace. When X is separable, every subspace of X is quasi-complemented.6' 7 Linden- strauss has recently shown4 that if r is uncountable, co(r) is not quasi-comple- mented in 1w(r). We sketch a proof below of the fact that co is quasi-comple- mented in 1X. This answers in the negative a question raised by Bade.' We also obtain that HI is quasi-complemented in Lw; in fact our results show that every weak* closed subspace of LX (with respect to LI), and every separable subspace of 1X, is quasi-complemented (cf. Corollary 3 and Theorem 8, below). All our results follow from the general criterion given in Theorem 1. Many of our applications of this criterion (cf. Corollaries 5, 6, and 7, and Theorems 8 and 9) require the notion of totally incomparable Banach spaces (defined just before Theorem 4). We also use this notion in Theorem 10, to obtain a partial converse to Theorem 1. We do not know if every subspace of 1- is quasi-complemented. It does follow from Theorem 1 that every subspace of 1- is isomorphic to a quasi-. Theorem 1, together with known results on Grothendieck spaces, implies that if A is a quasi-uncomplemented subspace of 1X, then the Banach space lw/A contains no separable quasi-complemented subspaces. We shall only sketch certain of the proofs here; detailed arguments and ex- tensions of these results will appear elsewhere. We wish to-thank W.' G. Bade for many stimulating conversations concerning this research. (2) Definitions and Notation.-We follow reference 3 for the most part. We recall briefly that if X is a Banach space, X* denotes its dual. If A c X or if Y C X*, A=I{feX*:f(a)=0 forall aeA}, y {x e X: y(x) =0 for all y eY}. X* is said to be weak* separable if there exists a countable subset of X*, dense in the weak* topology. (We remind the reader that if X is separable, then X* is weak* separable; and that every subspace of 1w has a weak* separable dual.) S denotes an infinite compact , C(S) the space of all (real or complex valued) continuous functions on S. X is called a Grothendieck space if weak* convergent in X* are weakly convergent. (We recall that 1w is a Grothendieck space.) Two Banach spaces are said to be isomorphic if there exists an invertible con- tinuous from one onto the other. 361 Downloaded by guest on September 24, 2021 362 MATHEMATICS: H. P. ROSENTHAL PROC. N. A. S.

If r is a set, I@(r) denotes the space of all bounded scalar-valued functions defined on r, co(F) its subspace of functions which are arbitrarily small off of finite sets. N denotes the natural numbers; I@(N) and co(N) are denoted by 1l and co, respectively. Finally, HO denotes the space of all uniformly bounded functions analytic in the open unit disk, regarded as a subspace of LX of the unit circle. (3) co Is Quasi-Complemented in 1X.-To sketch the proof, we first need the following preliminary observations: (i) A is quasi-complemented in X if and only if there exists a weak* closed subspace Y of X* such that Y n A = Y'nfA = {O}. (Indeed, if Y has these properties, then Y' is a quasi-complement for A.) (ii) If Y is a reflexive subspace of X*, then Y is weak* closed. (This follows from the Krein-Smulian theorem, and incidentally gives a linear topological characterization of reflexive spaces.) (iii) If S contains an infinite perfect subset, then there exists a subspace of C(S) * isomorphic to . (For then there exists a continuous mapping of a closed subset of S onto the unit interval; one then shows that LI [0, 1] is isometric to a subspace of C(S) *.) Now co0 may be identified with C(13N -- N)*, where AN denotes the Stone- Cech compactification of N. It follows from (iii) that there exists a subspace H, of col, isomorphic to Hilbert space. Let jh, MA2 ... be a basis for H,, equivalent to an orthonormal basis for Hilbert space, with |,nI = 1 for all n. For each n e N, let n,, be the on N such that a,, m I = bnm- Now let H be the -closed of { S./n + ru: n e N}. H is easily seen to be isomorphic to Hilbert space, hence H is weak* closed by (ii). Finally, H n co' = H' n co = {0 }, so H' is a quasi-complement for coby (i). (4) A General Criterion and Its Consequences.-From now on, we shall assume that A is a subspace of the Banach space X such that A* is weak* separable. THEOREM 1. If A' contains a reflexive subspace, then A is quasi-complemented. Theorem 1 is proved by expanding the ideas developed in Section 3- together with those in reference 2. Critical use is also made of Murray's result.' THEOREM 2. Let X = C(S), A of infinite codimension, and suppose that C(S)/A is isomorphic to a conjugate space. Then every subspace of A is quasi- complemented in X. Proof: If C(S)/A is reflexive, then so is its dual, A'. If C(S)/A is not re- flexive, then C(S)/A contains a subspace isomorphic to 1X.2, 5 Hence since 1O is injective, (lO)* is isomorphic to a subspace of (C(S)/A)*, so in any case, A' contains a reflexive subspace. COROLLARY 3. If B is a weak* closed subspace of 1w (with respect to 1) or of Lw (with respect to L1) of infinite codimension, then every subspace of B is quasi-com- plemented in 1w (resp. L@). The following concept enables us to develop quasi-complementation results based on the linear topological properties of A alone, and not on the way in which A is imbedded in X. Definition: The Banach spaces E and F are said to be totally incomparable if E and F have no isomorphic subspaces. Downloaded by guest on September 24, 2021 VOL. .59, 1968 MATHEMATICS: H. P. ROSENTHAL 363

THEOREM 4. If E and F are totally incomparable subspaces of X, then E' + F is norm-closed. The proof uses techniques similar to those of reference 2. We may assume that E n F = {IO}, since by hypothesis, E n F is finite dimensional. If E + F is not norm-closed, one may choose a basic (fj) in F, such that If"|I= 1 for all n, and such that lim [inf {If, - el: e eE, tell = 1}] = 0. It then follows that for some subsequence (fni), there exists a sequence (en) of ele- ments of E, such that (es) and (f,,) have isomorphic closed linear spans, a contra- diction. COROLLARY 5. Let Y be a subspace of X such that Y' contains no reflexive sub- spaces. Then every reflexive subspace of X* contains a finite-codimensional sub- space isomorphic to a subspace of Y*. We are able to show that if A is isomorphic to a subspace of LO [0, 1], then A* contains no isomorph of Ir if 1 < p < 2 and 1 < r < 2, or if

p- By using known results concerning the subspaces of LI [0, 1 ] and co ,we obtain COROLLARY 6. Let S have an infinite perfect subset, and X = C(S). If A is such that for some 1 < r < 2, no subspace of A* is isomorphic to L' [0,1], then A is quasi-complemented. In particular, if A is isomorphic to a subspace of co or of LO [0,1 for some 1 < p < co, then A is quasi-complemented. COROLLARY 7. Every C(S) contains a quasi-complemented, uncomplemented separable subspace. If S contains no infinite perfect subsets, the proof uses Mackey's result rather than Corollary 6. THEOREM 8. Every separable subspace of lO is quasi-complemented. To prove this, we observe that there exists a continuous map of 3N onto G, the Bohr compactification of the integers (cf. reference 9 for properties of the latter). Then we show that L'(G) is isometric to a subspace of (1O)*. Now the dual group of G is the circle group with the discrete topology, and the latter contains an independent subset r of cardinality c, the continuum. The closed linear span in L1(G) of the characters belonging to r (or of the real parts of these characters for the case of the real scalar field) is then contained in L2(G).9 It is thus established that (1O)* contains a subspace isomorphic to an insepa- rable Hilbert space. Now suppose that Y is a separable subspace of 1. Then any quotient of Y must be separable. Hence Y* cannot contain an isomorph of an inseparable Hilbert space, so by Corollary 5, Y' contains a reflexive subspace. THEOREM 9. Let r be a set of cardinality m, where 2m > c. Then if X = 1 (r), A is quasi-complemented. The proof is accomplished in a similar manner. In this case we obtain that (lw(F))* contains a subspace isomorphic to a Hilbert space of dimension 2m. Downloaded by guest on September 24, 2021 364 MA THEMA TICS: H. P. ROSENTHAL PROC. N. A. S.

Since A* is assumed weak* separable, A has cardinality at most the continuum, so again A' contains a reflexive subspace by Corollary 5. We wish finally to state a partial converse to Theorem 1. We were led to this result by some recent work of Lindenstrauss. THEOREM 10. Let Y be a subspace of C(S), where C(S) is a Grothendieck space. Suppose that the unit ball of Y* is weak* sequentially compact. Then if Y is quasi-complemented, Y' contains a reflexive subspace. Proof: Suppose that B is a quasi-complement. It then follows that the unit ball of B' is weakly sequentially compact, hence B' is reflexive. Now if Y' contains no reflexive subspace, then Y' + B' is norm-closed, by Theorem 4. By a result of Reiter's,8 Y + B is then norm-closed. Hence Y + B = C(S). So Y is complemented, and Y* is isomorphic to B'. But then Y is reflexive, a contradiction, since no reflexive subspace of C(S) is complemented.3 * The research for this paper was partially supported by National Science Foundation grant NSF-GP-5585. 1 Bade, W. G., "Extensions of interpolation sets," in , Proceedings of the Irvine Conference (Thompson Book Co., 1967). 2 Bessaga, C., and A. Pelczynski, "On bases and unconditional convergence of series in Banach spaces," Studia Math., 17, 151-164 (1958). 3 Dunford, N., and J. T. Schwartz, Linear Operators: Part I, General Theory (New York: Interscience Publishers, 1958). 4Lindenstrauss, J., "On subspaces of Banach spaces without quasicomplements," Israel J. Math., to appear. 6 Pelcznski, A., "Projections in certain Banach spaces," Studia Math., 19, 209-228 (1960). 6 Mackey, G., "Note on a theorem of Murray," Bull. Am. Math. Soc., 52, 322-325 (1946). 7 Murray, F. J., "Quasi-complements and closed projections in reflexive Banach spaces," Trans. Am. Math. Soc., 58, 77-95 (1945). 8 Reiter, H. J., "Contributions to harmonic analysis, VI," Ann. Math., 77, 552-562 (1963). 9 Rudin, W., Fourier Analysis on Groups (New York: Interscience Publishers, 1962). Downloaded by guest on September 24, 2021