ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X III (1983) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE X X III (1983) Marek W ôjtowicz (Poznan) On the James spaceJ(X ) for a Banach space X Abstract. For X an arbitrary Banach space, we consider the natural analogue J (X) of the famous James spaceJ — J (В ). Our main result is that J (X)** is isomor­ phic to J(X**)@X**. In particular, if X is reflexive, thenJ (X)**/J(X) is isomorphic to X. The Banach space constructed by В. C. James [3] in 1950, now com­ monly denoted by J , was the first example of a quasi-reflexive space (i.e. of finite codimension in its bidual) of order 1 : dim = 1 . In 1971, J . Lindenstrauss [4] (see also [5], Theorem l.d.3) modified James, construction so as to prove that for every separable Banach space X there exists a Banach space Z such that Z**\Z is isomorphic to X. The same holds for weakly compactly generated spaces X , as shown by Davis, Figiel, Johnson and Felczynski [ 1] in 1974; this class contains all separable and all quasi-reflexive Banach spaces. It seems to be an open question whether any restrictions on X are necessary at all. Our result thus shows that for X a reflexive Banach space we may take Z = J (X) to have Z**/Z isomor­ phic to X. Our terminology and notation are mostly standard (e.g., as in [5]). We write X ^ Y or X Y if the Banach spaces X and Г are isometric or isomorphic, respectively. Let Y — (Y, ||-||) be a Banach space, Г a finite set and l< p < oo. Then 1р(Г, Y) denotes the Banach space consisting of all families у — (уу)уеГ with yY e T, under the norm ||-||р defined by the formula \\y\\p = { V*r if p — oo. We have under a natural isometry (cf. [ 2 ], Chap. П, §2, (11)). 184 M. Wôjtowicz Let X = (X, ||*||) be a Banach space. Then the Banach space J (X ) = (J(X), 111*111) is defined as follows: J(X ) = {x = (х{) e X N: |||æ|j| = sup<rp(®) < oo}, where & is the set of all strictly increasing sequences p = (^(1), ... ...,р (2 Ь +1)) in N ,keN u {0}, and к Gp{pC) ( ll*^p(2f) Xp{2i— l) ll2 “b 11^(2*4-1)11 j * t’ = l It is easily seen that for each x = (x{) e J (X) the limit x0 = lim^ exists г->оо in X. The James space J (X) = ( J (X)] |||*|||) is the closed subspace of J (X) consisting of those x e J (X) for which lim xi = 0. It is clear that the sub- i-*r OO space X of all constant sequences is isometric in a natural manner to X and is the range of the norm 1 projection Q in J (X) defined by Q{x) = (x0, x0, ...). Moreover, Q~1( 0) = J (X). Hence (1) J(X)=X@J(X). Consider the space X x J (X) equipped with the norm Ц-lb defined by IK^oj ®)lli = ЩН-HI N11» and define T: X x J(X)->J{X) by T(xQ, x) — (х0) #?i, x2y...). Then T 'is an algebraic isomorphism between XxJ(X) and J(X), and ||(a?0, a?) ||x < 2 111 T(x0, ж)|||. Hence T~l is continuous, and so is Pby theBanach isomorphism theorem (this can also be checked directly). Thus T is a topo­ logical isomorphism between XxJ(X) and J(X), whence XxJ(X)^J{X). Hence and from (1) we have (2) J(X )^J(X ). For each n e N the map P n: x ь-> (0, ..., 0, xn, 0, ...) is a projection in CO J (X) of norm 1. Moreover, PnPm — 0 if n Ф m, and x = 2 -P»®* Y xeJ{X). 71—1 This means that the sub spaces Pn(j (X)), n = 1, 2, ..., form a Schauder decomposition of J (X), with P 1? P 2,... as the associated sequence of projections. Write = -P1+-P2+ ••• J n(X) = Sn(J(X)) = {CO = (xJeJiX): x{ = 0 Vi > »}, James spaceJ(X) for a Banach space X 185 and observe that 8n is a projection in J (X) and \\SJ=1 (neN). Similarly as in the case of the usual James space J — J (JR) (cf. [3]), it can be shown that the just described decompo sition of J (X) is shrinking, i.e., for every f e J(X)*, * OO (3) / — J ? P * f = lim 8 * f (norm convergence), n~=\ oo so that thé sequence of subspaces P*(J (X)*), n = 1, 2, ..., forms a Schau- der decomposition of J (X)*, with P *, P *, ... as the associated sequence of projections. Since X and P n[ J (X)) are isometric via the natural injection, for every / e J (X)* there is a unique f n e X* such that (Kf)(x) = VæeJ(X). We may thus identify every / e J (X)* with the corresponding sequence (/») c X*; thus ' OO f(x) = ^fn(Xn), VxeJ(X). n~l Similarly, for every F eJ(X)** there is a unique sequence F = (Fn) in X ** such that for all n e N and / = (fn) e J (X)*, OO (PTF)(f)-Fn(fn) and F(f) = n—1 We shall now prove our main result. T h e o r e m . The map Fi->F = (Fn) is an isometry between J(X )** and J(X**). Proof. For each p = (p( 1), p(2k + l)) e& consider the operator Bp: J(X)->lz(X) defined by B p (%>) = (%p{2) •••? ®р(2к) ^р(2к— 1)? ^p(2k-\-lp •*•)• Then * H-BpMII = ap (x) <|||®|||, and hence |]Бр || = 1. It is easily seen that B** : J(X )**-*l2(X)** = l2(X**) is given by Bp*(F) = (Fp(2)—Fp(i)i ••• / Fp(2k)~Fp(2k-i)i —F P(2k+i)i h, • ••)• Hence \\b ;*(F )\ \ =op (é ), 186 M. Wôjtowicz and since ||J8j*|| = ||2?p || = 1, we have , ■/ ap (F)^\\F\\. It follows that (4) l!|#|i|<m, V F e J ( X Г. The proof of the converse inequality requires some preparations. For each n e N let 3?n be the set of those p e g ? whose last term is < n - f 1. Define ^ n — ^oo { ^ n l ^2 ( - ^ ) ) and note that (under a natural isometry). It is clear that the map An: J n(X)-~+3Fn defined by A n ( æ ) = ( B p i æ ) ) ^ is am isometric embedding. This and the Hahn-Banach theorem imply that for every y> e J n (X)* there is g = {gp)pe£?n e 3C*n such that Ml = Ml = 2 IM P&n and y> = A*g, i.e., v “ (BP\jn(x)f (gP) • P ^ n In particular, if / e J(X)* and y = (S*f)\Jn{X), then ||y|| = \\S*J\\ and so we have (5) . iis:/n - 2" w and s«f = 2 Biop- p e& n p e & n Fix F e J (X )** and let/ e J (X)*. Then F(f) = lim F(8*f) by (3), and hence n-yoo (6) \F(f)\ < sup \F(8*f)\. П Using (6), we get •p(«:/) = Z F (Ku,>) = S (K *F)(sP), p e P n P e &n Jâmes space J (X) for a Banach space X 187 and hence |.F(Slft|<m ax||£;*F||- У Ifell = m ax ||В**Л|• | Ю Pe^n < sup ||B” B||-Ii/I| = infill • ll/il. p e # In view of (6) we have thus shown that Ж infill. Combined with (4) this proves {7) ' 111*111 = P 4I< oo, VF e J (X )* * . Now suppose Ф = (Fn) e J (X**), i.e., |||Ф|Ц < oo. For each n eN let Фп = (F1,...,Fn, 0, ...) and define FneJ(X)** by П F(f) = ^ F A fi), У/ = (/<) 6J( I ) | t=l elearly, F 11 = Фп, and so \\Fn\\ = |||ФП||| < Ц|Ф||| by (7). Moreover, Fn(f) — Fn(8*f), and if m < n, then n I 2 w t)I = i^‘(«„*/)--p“( O i < г = т + 1 < 11|Ф|1НЙ/-£т/1К0 as m,n-+oo by (3). Therefore the formula OO F(}) = lim *"*(/) = f = (/«) e J(X f, n-*0О t-=l defines a linear functional F on J (X)*, which is continuous, because |J?(/)K sup|i"(/)|< 111ФЦ1-11Я- П Clearly Ф — F , and this completes the proof of the theorem. Using (1) and ( 2 ) for X** instead of X , we now have COEOLLAEY 1. J(Xf* 9* J(X**) = J(X**)@X** J(X**). Coeollaey 2. I f X is reflexive, then J(X )** g*J{X) &J(X); hence J(X)**/J(X) ъ X. Eemarks. (1) If X Ф {0}, then the natural decomposition of J(X) is not boundedly complete (by an argument similar to that used for J, cf. 188 M. Wojtowicz [3]), and hence J (X) is not reflexive. This conclusion is based on the result of Sanders [ 6] stating that a Banach space Z with a Schauder decomposi­ tion (Zn) is reflexive iff the decomposition is shrinking and boundcdly complete and all Zn are reflexive. (2) Corollary 2 for more general James-Orliez spaces has been recently announced by P.Y. Semenov [7]. (The results of the present paper were obtained independently.) The author wishes to thank Professor L. Drewnowski for his valuable assistance and helpful suggestions. References [1 ] W. J. Davis, T. Figiel, W. B. Johnson and A. Pelczyhski,Factoring weakly compact operators, J.