The descriptive complexity of the family of Banach spaces with the πproperty


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1 Arab. J. Math. (2015) 4:35 39 DOI /s Araban Journal of Mathematcs Ghadeer Ghawadrah The descrptve complexty of the famly of Banach spaces wth the πproperty Receved: 25 March 2014 / Accepted: 4 August 2014 / Publshed onlne: 27 August 2014 The Author(s) Ths artcle s publshed wth open access at Sprngerlnk.com Abstract We show that the set of all separable Banach spaces that have the πproperty s a Borel subset of the set of all closed subspaces of C( ), where s the Cantor set, equpped wth the standard EffrosBorel structure. We show that f α<ω 1, the set of spaces wth Szlenk ndex at most α whch have a shrnkng FDD s Borel. Mathematcs Subject Classfcaton 46B20 54H05 1 Introducton Let C( ) be the space of contnuous functons on the Cantor space.itswellknownthatc( ) s sometrcally unversal for all separable Banach spaces. We denote SE the set of all closed subspaces of C( ) equpped wth the standard EffrosBorel structure. In [1], Bossard consdered the topologcal complexty of the somorphsm relaton and of many subsets of SE. In addton, t has been shown that the set of all separable Banach spaces that have the bounded approxmaton property (BAP) s a Borel subset of SE, and that the set of all separable Banach spaces that have the metrc approxmaton property (MAP) s also Borel [5]. We recall that the Banach space X has the π λ property f there s a net of fnte rank projectons (S α ) on X convergng strongly to the dentty on X wth lmsup α S α λ (see [2]). We say that the Banach space X has the πproperty f t has the π λ property for some λ 1. In ths note we show that the set of all separable Banach spaces that have the πproperty s a Borel subset of SE. Ths bears some consequences on the complexty of the class of spaces wth fntedmensonal decompostons. For nstance, we show that n the set of spaces whose Szlenk ndex s bounded by some countable ordnal, the subset consstng of spaces whch have a shrnkng fntedmensonal decomposton s Borel. G. Ghawadrah (B) Unversté Pars VI Case 247, 4 Place Jusseu, Pars Cedex 05, France Emal:
2 36 Arab. J. Math. (2015) 4: Manresults Here s the man techncal lemma. Lemma 2.1 Suppose (x n ) n=1 s a dense sequence n a Banach space X. Then, X has the πproperty f and only f λ >1 c (0, 1 4 ) Q K ɛ >0 λ >λ R N R σ 1,...,σ N Q R α 1,...,α N Q R α σ ( j)x j λ α x. (2.1) K, R x σ ( j)x j ɛ (2.2) α R σ ( j)x j α R R σ ( j)σ j (t) x t c α x. (2.3) t=1 where K, R, N vary over N and ɛ, λ, and λ vary over Q. Proof Indeed, suppose X has the π λ property. Then there exsts a sequence (P n ) of fnte rank projectons such that P n <λ,foralln and P n converge strongly to the dentty. By perturbng P n, we may suppose that P n maps nto the fntedmensonal subspace [x 1,...,x Rn ] for some R n n N but we stll have (2.2)and P n <λ. Then, for every N, we may perturb P n slghtly so that P n <λ and P n (x ) belongs to the Qlnear span of the x j for all N, such that (2.1), (2.2)and(2.3) stll hold. Defne now (σ (n) ) (Q R n) N, such that P n (x ) = R n σ (n) ( j)x j.sncep 2 n (x ) = R n three nequaltes hold for all α 1,...,α N Q, and K. Conversely, suppose that the above crteron holds and that ɛ > 0. Pck a ratonal a K.Soletλ and R be gven as above. Then for every N and N, defney N z N t=1 [ Rn σ ( j)σ j (t) ɛ 3λ ] x t,the >ɛ>0and = R σ ( j)x j,and = [ R R ] t=1 σ ( j)σ j (t) x t n [x 1,...x R ], where the σ are gven dependng on N.Wehavethat α y N λ α x. (2.4) for all α Q, x y N ɛ (2.5) for all K,and α y N α z N c α x. (2.6) for all c (0, 1 4 ) Q. In partcular, for every, the sequences (y N ) N=,and(z N ) N= are contaned n a bounded set n a fntedmensonal space. So by a dagonal procedure, we may fnd some subsequence (N l ) so that y = lm y N l l and z = lm z N l l exsts for all. By consequence α y λ α x. (2.7)
3 Arab. J. Math. (2015) 4: for all α Q, x y ɛ (2.8) for all K,and α y α z c α x. (2.9) for all c (0, 4 1 ) Q. Now, snce the x are dense n X, there are unquely defned bounded lnear operators T K,ɛ : X [x 1,...,x R ] satsfyng T K,ɛ (x ) = y and then TK,ɛ 2 : X [x 1,...,x R ] satsfes TK,ɛ 2 (x ) = z such that T K,ɛ λ <λand x T K,ɛ (x ) ɛ for all K.LetK and ɛ 0. Then T K,ɛ (x ) x for all x (x ) strongly. Snce (x ) s a dense sequence n X and the operators T K,ɛ are unformly bounded, then T K,ɛ (x) x for all x X strongly. Also, lm sup T K,ɛ TK,ɛ 2 =c < 1 4. Therefore, by [3, Theorem(3.7)], X has the π λ+1 property as c 0. Theorem 2.2 The set of all separable Banach spaces that have the πproperty s a Borel subset of SE. Proof Let K, R, N vary over N and ɛ, λ vary over Q. Let also c (0, 1 4 ) Q, σ (Q R ) N,andα Q N. Then we consder the set E c,k,ɛ,λ,r,n,σ,α C( ) N such that: E c,k,ɛ,λ,r,n,σ,α ={(x n ) n=1 C( )N ; (2.1) (2.2) and (2.3) hold} Ths set s closed n C( ) N. Therefore, for λ R E λ = c ɛ K λ >λ R N σ (Q R ) N α Q N E c,k,ɛ,λ,r,n,σ,α s a Borel subset of C( ) N. Moreover, the set E = λ QE λ s also Borel. There s a Borel map d : SE C( ) N such that d(x) = X, by[8, Theorem (12.13)]. Moreover, the prevous Lemma mples that X has the πproperty d(x) E. Therefore, {X SE; X has the πproperty} s a Borel subset of SE. We wll now prove that ths result mples, wth some work, that n some natural classes the exstence of a fntedmensonal decomposton happens to be a Borel condton. We frst consder the class of reflexve spaces. The commutng bounded approxmaton property (CBAP) mples the bounded approxmaton property (BAP) by the defnton of the CBAP. By Grothendeck s theorem (see [9, Theorem 1.e.15]), the BAP and the metrc approxmaton property (MAP) are equvalent for reflexve Banach spaces. In addton, [3, Theorem 2.4] mples that for any reflexve Banach space the CBAP s equvalent to MAP. For the set R of all separable reflexve Banach spaces, Theorem (2.2), [5, Theorem2.2] and [2, Theorem 6.3] mply that there exsts a Borel subset B ={X SE; X has the MAP and the πproperty} such that {X R; X has a FDD} =B R. We wll extend ths smple observaton to some classes of nonreflexve spaces. The followng result has been proved n [7]. The proof below follows the lnes of [6]. Proposton 2.3 Let X be a Banach space wth separable dual. If X has the MAP for all equvalent norms, then X has the MAP.
4 38 Arab. J. Math. (2015) 4:35 39 Proof Snce X s separable, there s an equvalent Fréchet dfferentable norm on X. If. X s a Fréchet dfferentable norm and x S X, there exsts a unque x S X such that x (x) = 1, and x s a strongly exposed pont of B X. Snce by assumpton X equpped wth ths norm has the MAP, there exsts an approxmatng sequence (T n ) wth T n 1, and then for all x X we have Tn (x ) w x.forallx X whch attans ts norm, we have Tn (x ) x X 0. Bshop Phelps theorem yelds that for all x X, Tn (x ) x X 0. Therefore, X has the MAP. The set SD of all Banach spaces wth separable dual spaces s coanalytc n SE and the Szlenk ndex Sz s a coanalytc rank on SD (see [1, Corollary (3.3) and Theorem (4.11)]). In partcular, the set S α ={X SE; Sz(X) α} s Borel n SE (see [8]). In ths Borel set, the followng holds. Theorem 2.4 The set of all separable Banach spaces n S α that have a shrnkng FDD s Borel n SE. Proof Indeed, by [4, Theorem 1], we have that S α ={Y SE; X S α wth Y X } s analytc. Snce the set {Y SE; Y has the BAP} s Borel by [5, Theorem 2.2], then G α ={Y SE; X S α wth Y X and Y has the BAP} s analytc. By [4, Proposton 7], we have that G α ={X S α ; Y G α, wth Y X } s analytc. Snce {(X, Z); Z X} s analytc n SE SE ([1, Theorem 2.3]) and {Z; Z fals the MAP} s Borel [5], the set {(X, Z); Z X, Z fals the MAP} s analytc, thus ts canoncal projecton {X SE; Z SE; Z X, Z fals the MAP} s analytc. Now, Proposton (2.3) mples that the set H α ={X S α ; X fals the AP} ={X S α ; Z wth Z X and Z fals the MAP} s analytc. Snce S α \ H α = G α and both G α and H α are analytc sets n SE, then both are Borel by the separaton theorem. Now, [2, Theorem 4.9] mples that G α ={X S α ; X has the shrnkng CBAP}. Thus, {X S α ; X has a shrnkng FDD} s Borel by Theorem (2.2)and[2, Theorem 6.3]. Questons: As seen before, a separable Banach space has CBAP f and only f t has an equvalent norm for whch t has MAP. It follows that the set {X SE; X has the CBAP} s analytc. It s not clear f t s Borel or not. Also, t s not known f there s a Borel subset B of SE, such that {X SD; X has the AP} =B SD. Ths would be an mprovement of Theorem (2.3). Fnally, what happens when we replace FDD by bass s not clear: for nstance, the set of all spaces n S α whch have a bass s clearly analytc. Is t Borel? Acknowledgments I would lke to thank my Ph.D. supervsor Glles Godefroy at Unversty of Pars VI for hs suggestons and encouragement, and the referee for a number of helpful suggestons for mprovement n the artcle. Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton Lcense whch permts any use, dstrbuton, and reproducton n any medum, provded the orgnal author(s) and the source are credted.
5 Arab. J. Math. (2015) 4: References 1. Bossard, B.: A codng of separable Banach spaces, analytc and coanalytc famles of Banach spaces. Fund. Math. 172, (2002) 2. Casazza, P.G.: Approxmaton propertes. Handb. Geom. Banach Spaces 1, (2001) 3. Casazza, P.G.; Kalton, N.J.: Notes on approxmaton propertes n separable Banach spaces. Lect. Notes Lond. Math. Soc. 158, (1991) 4. Dodos, P.: Defnablty under dualty. Houst. J. Math. 36(3), (2010) 5. Ghawadrah, G.: The descrptve complexty of the famly of Banach spaces wth the bounded approxmaton property. Houst. J. Math. (to appear) (2014) 6. Godefroy, G.; Saphar, P.: Dualty n spaces of operators and smooth norms on Banach spaces. Ill. J. Math. 32(4), (1988) 7. Johnson, W.B.: A complementably unversal conjugate Banach space and ts relaton to the approxmaton problem Isr. J. Math. 13, (1972) 8. Kechrs, A.S.: Classcal Descrptve Set Theory. Sprnger, Berln (1995) 9. Lndenstrauss, J.; Tzafrr L.: Classcal Banach Spaces I. Sprnger, Berln (1977)
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