[Math.FA] 20 Sep 2000 That Hilbertian of K Eoeby Denote Ec Also Hence San Is Ibr Pcsejyteapoiainpoet [P])

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AND IA, = j j ( ota every that so N X ,K m, ota o every for that so 2 1 P i , satisfies = X n ∞ 1 = o all for 0 = ) =0 { X j has | , ℓ p p k ,w aethat have we 2, ssi obe to said is n n n p 2 H H k n o 010366-163. No. +1 2 n ( and ) ( ( m oet.Moreover, roperty. ,m K m, n, with ,m K m, n, ≤ dmninlsubspace -dimensional tDS9220 DMS- DMS-9623260, nt to property. ation m stedirect the as n m p < j p h rwhrate growth The . ce k n hr exists there ie integers Given . n rvddthere provided ) .Tu if Thus ). ↓ E eso that show we ) asymptotically n and 2 which 2( Z rinspace ertian k a Hilbert eak +1) ox- = oHilbert to inprop- tion s k , Z k n 1 Z ≥ ⊕ ∞ ↑ X n tian, and Z 0 so is } 2 . 2 P. G. CASAZZA, C. L. GARCÍA, AND W. B. JOHNSON of HX (m,K) for a fixed K as m is one measurement of the closeness of X to a Hilbert space. → ∞ A Banach space is called a weak Hilbert space provided that there are positive constants δ and K so that for every n, every n dimensional subspace of X contains a further subspace E of dimension at least δn so that E is K-isomorphic to a Hilbert space and E is K-complemented in X (that is, there is a projection having norm at most K from X onto E). The definition of the property H(n,m,K) was made in [J], although the nomen- clature “asymptotically Hilbertian” was coined by Pisier [P]. Weak Hilbert spaces were introduced by Pisier [P], who gave many equivalences to the property of being weak Hilbert; we chose the one most relevant for this paper as the definition of weak Hilbert. Relations between the weak Hilbert property and the asymptotically Hilbertian property are given in [J] and [P]. First, a weak Hilbert space must be asymptotically Hilbertian ([P, Section 4]). It seems likely that if X is weak Hilbert then for some K, HX (m,K) Km, but in fact no reasonable estimates are known for HX (m,K) when X is a weak≤ Hilbert space. It is known (see [CJT], [NT-J]) that if X is a weak Hilbert space which has an unconditional basis and there is a K so that HX (m,K) is dominated by f(m) for some iterate f of exp, then for any iterate g of log there is another constant K′ so that H (m,K′) K′g(m). In the other direction, it X ≤ follows from [J] that if for some K the sequence HX (m,K) grows sufficiently slowly as m (say, like log log m), then X is a weak Hilbert space. In this paper we are interested→ ∞ in examples of spaces which are of type 2. We refer to Chapter 11 of [DJT] or Section 1.4 of [T-J] for the definitions and basic theory of type p and cotype p as well and the type p and cotype p constants Tp(X), Cp(X) of a Banach space X. Relevant for us is that if X is a type 2 space and E is a subspace of X which is K-isomorphic to a Hilbert space then, by Maurey’s extension theorem, E is T2(X)K- complemented in X ([DJT, Corollary 12.24]). Thus it is clear that if X is of type 2 and for some K, H (m,K) Km, then X is weak Hilbert. Here X ≤ we should mention that by [FLM], polynomial growth of HX (m,K) (as m ) ′ ′ → ∞ implies linear growth of HX (m,K ) for some K . Our main interest here is the linkage among the weak Hilbert property, the asymptotically Hilbertian property, and the approximation property. The argu- ments in [J] show that if X has type 2 and for some K, HX (m,K) K log m for infinitely many m, then all subspaces of X (even all subspaces of≤ every quotient of X) have the approximation property. In [P] it is shown that all weak Hilbert spaces have the approximation property. Thus if X is of type 2 and for some K, HX (m,K) Km for all m, then all subspaces of every quotient of X have the approximation≤ property. It is easy to build examples of a type 2 space X for which there is a constant K so that for any iterate f of the log function, HX (m,K) Kf(m) for infinitely many m and yet X is not a weak Hilbert space. Now such a≤ space X is in some sense close to Hilbert space and, in particular, every subspace of X has the approximation property. In this paper we show that there are two such spaces; call them Z1 and Z2; so that Z := Z1 Z2 has a subspace which fails the approximation property. Moreover, Z has an⊕ unconditional basis ∞ (Z = ℓkn for appropriate p 2 and k ) and is nearly a weak Hilbert n=0 pn n ↓ n ↑ ∞ spaceP in the sense that for some K, the growth rate of HZ (m,K) as m is log log m → ∞ close to being polynomial in m (HZ (m,K) m is what we get; recall that ≤ ′ ′ polynomial growth of HX (m,K) gives linear growth of HX (m,K ) for some K .). AN ASYMPTOTICALLY HILBERTIAN SPACE WHICH FAILS THE A.P. 3

2. the example We follow closely A. M. Davie’s construction of a Banach space failing the approximation property [D]. Davie constructed for p> 2 a subspace of ℓ = ℓkn p p p kn which fails the approximation property. He could as well have used ℓPp for ∞ r any 1 r . Here we use instead Z := ℓkn where p 2 appropriatelyP ≤ ≤ ∞ n=0 pn 2 n ↓ and kn as in [D]. Basically we compute howP fast pn can go to 2 so that Davie’s argument yields a subspace of Z which fails the a.p.. The obvious condition is 1/2−1/p 1/2−1/p ∞ that k n cannot be bounded, for if k n is bounded then ℓkn n n n=0 pn 2 is isomorphic to ℓ2. P n For any integer n 0 consider an Abelian group Gn of order kn = 3 2 , and n n n ≥ n · let σ1 ,... ,σ2n , τ1 ,... ,τ2n+1 be the characters of Gn. Lemma (b) in [D] shows that this enumeration of the characters of Gn can be chosen so that there exists an absolute constant A> 0 such that for all g G , ∈ n

2n 2n+1 (2.1) 2 σn(g) τ n(g) A(n + 1)1/22n/2. | j − j |≤ Xj=1 Xj=1 n Let G be the disjoint union of the sets Gn and for each n 0 and 1 j 2 deﬁne en : G C via: ≥ ≤ ≤ j →

n−1 τ (g), if g G − ,n 1 j ∈ n 1 ≥ en(g)=  εnσn(g), if g G j j j ∈ n  0, otherwise n  where εj = 1 is a choice of signs for which the inequality (2.5) below is satisﬁed. ± n ∞ To deﬁne E let, as above, kn =3 2 and let (pn)n=0,2

n n Deﬁne E to be the closed linear span in Z of ej n 0, 1 j 2 . To show that E fails the approximation property one proceeds{ | as≥ follows:≤ ≤ } For n 0 and 1 j 2n deﬁne αn E∗ by ≥ ≤ ≤ j ∈

n −1 −n n n −1 (2.2) αj (f)=3 2 εj σj (g )f(g). gX∈Gn When n 1 the expression above equals ≥

n −1 1−n n−1 −1 (2.3) αj (f)=3 2 τj (g )f(g). g∈XGn−1 This follows from the fact that αn(ek)= δ δ (because of the orthogonality j i ij · kn of the characters of a group) and then a linearity and continuity argument shows that (2.2) and (2.3) agree on E. 4 P. G. CASAZZA, C. L. GARC´IA, AND W. B. JOHNSON

Now let B(E) be the space of bounded, linear operators on E, and for each n 0 deﬁne βn in the dual space B(E)∗ as: ≥

2n βn(T )=2−n αn(T (en)),T B(E) j j ∈ nX=1 Using (2.2) we can rewrite βn as:

2n βn(T )=3−14−n T εnσn(g−1)en (g) j j j gX∈Gn Xj=1 and from (2.3) we get:

2n+1 βn+1(T )=6−14−n T τ n(g−1)en+1 (g) j j gX∈Gn Xj=1 hence,

(2.4) βn+1(T ) βn(T )=3−12−n T (Φn)(g) − g gX∈Gn where,

2n+1 2n Φn =2−n−1 τ n(g−1)en+1 2−n εnσn(g−1)en, g G g j j − j j j ∈ n Xj=1 Xj=1 Note that Φn E for every g G and n 1. Now we estimate the right hand g ∈ ∈ n ≥ side of (2.4). If n 1 and g G then, ≥ ∈ n

−1 −n n n n 3 2 T (Φg )(g) sup T (Φg ) ∞ sup T (Φg ) Z . | |≤ g∈Gn{k k }≤ g∈Gn{k k } gX∈Gn Therefore,

n+1 n n β (T ) β (T ) sup T (Φg ) Z for every T B(E). | − |≤ g∈Gn{k k } ∈ Note that from (2.1) we have that Φn(h) A(n + 1)1/22−n/2 for g,h G . By | g |≤ ∈ n applying lemma (a) in [D], the signs εn, 1 j 2n, can be chosen so that j ≤ ≤

n 1/2 −n/2 (2.5) Φ (h) A (n + 1) 2 for g G ,h G − (n 1) | g |≤ 2 ∈ n ∈ n 1 ≥ where A2 is some absolute constant. An algebraic argument shows that a similar estimate can be obtained for g Gn and h Gn+1. In brief, we have that there is an absolute constant, say A, such∈ that, ∈

n 1/2 −n/2 (2.6) Φ (h) A(n + 1) 2 for g G and h G − G G . | g |≤ ∈ n ∈ n 1 ⊔ n ⊔ n+1 Now, if n 1 and g G then, ≥ ∈ n AN ASYMPTOTICALLY HILBERTIAN SPACE WHICH FAILS THE A.P. 5

n+1 2/pj Φn 2 = Φn(h) pj k g kZ | g | j=Xn−1 hX∈Gj − − A2(n + 1)2 n (3 2n 1)2/pn−1 + (3 2n)2/pn + (3 2n+1)2/pn+1 ≤ · · · − − 3A2(n + 1)2 n 22(n 1)/pn−1 + 22n/pn + 22(n+1)/pn+1 ≤ − 18A2(n + 1)22n(1/pn+1 1/2) ≤ thus,

− (2.7) Φn 3√2A(n + 1)1/22n(1/pn+1 1/2) k g kZ ≤ Consider the set = e0 (n + 1)2Φn g G ,n 1 C { 1}∪{ g | ∈ n ≥ } The estimate in (2.7) clearly shows that when − (2.8) (n + 1)5/22n(1/pn+1 1/2) 0 → the set becomes a relatively compact subset of E. Obviously there are many C ∞ α choices for (pn)n=0, pn 2, that satisfy (2.8); in particular, 1/pn =1/2 1/(n+1) for any α< 1 gives a sequence↓ satisfying (2.8). When n(1/p 1/2) =− 3 log (n+ n+1− − 2 1), the sequence (pn) is the one with the slowest (up to a constant) possible rate of decrease for this construction. This makes the space Z “almost” a weak Hilbert log log m space in the sense that for some K, HZ (m,K) m for large m. Indeed, ∞ kj ≤ consider F , a subspace of j=n+1 ℓpj of dimension m = m(k1 + + kn), 2 ··· P where m is the largest integer such that 0 < 1/2 1/pn+1 < 1/ log2(m). Then, d(F,ℓm) T (F )C (F ). The type 2 constant of−F is bounded by an absolute 2 ≤ 2 2 constant independent of m, say c1. The cotype 2 constant of F can be estimated, using Tomczak’s lemma, by the cotype 2 constant C2,m( ) on m vectors (see Section 5.25 in [T-J]): ·

− C (F ) √2C (F ) √2C (F )m1/2 1/pn+1 2 ≤ 2,m ≤ pn+1 − √2c m1/2 1/pn+1 ≤ 2 2√2c (by the choice of m). ≤ 2 m Hence, for K := 2√2c1c2 we obtain that d(F,ℓ2 ) K. Finally, to show that E fails the approximation property≤ the argument in [D] ﬁnishes as follows: for every T B(E), ∈

n+1 n n −2 β (T ) β (T ) sup T (Φg ) Z (n + 1) sup T x Z | − |≤ g∈Gn{k k }≤ x∈C k k Also, 0 1 β (T ) Te0 sup T x Z | |≤k k≤ x∈C k k n Hence β(T ) = lim →∞ β (T ) exists for all T B(E) and satisﬁes n ∈ β(T ) 3 sup T x Z | |≤ x∈C k k 6 P. G. CASAZZA, C. L. GARC´IA, AND W. B. JOHNSON

In particular, when is compact, β is a continuous linear functional on B(E) when B(E) is given theC topology of uniform convergence on compact sets. n n If IE is the identity map on E, it follows from the deﬁnition of β that β (IE )=1 for all n, so β(IE ) = 1. On the other hand it is easy to see that β vanishes on the set of ﬁnite rank operators on E, thus E cannot have the approximation property. Remark . ∞ 2.1 For (pn, kn)n=0 as above, we obtained an asymptotically Hilbertian space Z which has a subspace failing the approximation property. The space Z can be decomposed as the direct sum of two subspaces, say Z1 and Z2, all of whose subspaces have the approximation property. Indeed, as in example 1.g.7 in [LT], it is enough to construct a subsequence (pnj ) of (pn) as follows: set pn1 = p0 and kn1 = k0. Having chosen pn1 pnj (and their respective kn1 knj ), choose ··· ··· j kn pn such that if F ℓp (2

References

[CJT] P. G. Casazza, W. B. Johnson, L. Tzafriri, On Tsirelson’s space, Israel J. Math. 47 (1984), 81-98. [D] A.M.Davie, The approximation problem for Banach spaces, Bull. London Math. Soc. 5 (1973), 261-266. [DJT] J. Diestel, H. Jarchow, A. Tonge, Absolutely summing operators, Cambridge University Press, Cambridge, 1995. [FLM] T. Figiel, J. Lindenstrauss, V.D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), 53-94. [J] W. B. Johnson, Banach spaces all whose subspaces have the approximation property, Special Topics of Applied Mathematics, North–Holland (1980), 15–26 (also in S´eminaire d’Analyse Fonctionelle 79/80, Ecole Polytechnique. Palaiseau. Expos´eNo. 16). [LT] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, Vol. II, Springer Verlag, New York, 1979. [M] V.Mascioni, On Banach spaces isomorphic to their duals, Houston J. Math. 19 (1993), 27-38. AN ASYMPTOTICALLY HILBERTIAN SPACE WHICH FAILS THE A.P. 7

[NT-J] N. J. Nielsen, N. Tomczak-Jaegermann, Banach lattices with property (H) and weak Hilbert spaces, Illinois J. Math. 36 (1992), 345-371. [P] G.Pisier, Weak Hilbert spaces, Proc. London Math. Soc. 56 (1988), 547-579. [S] S. Szarek, A Banach space without a basis which has the bounded approximation property, Acta Math. 159 (1987), 81-98. [T-J] N. Tomczak-Jaegermann, Banach-Mazur distances and ﬁnite-dimensional operator ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics 38, Longman (1989).

Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65211 U.S.A. E-mail address: [email protected]

Department of Mathematics, Texas A&M University College Station, TX 77843–3368 U.S.A E-mail address: [email protected]