Continuous projections onto ideal convergent sequences
aa r X i v : . [ m a t h . F A ] N ov CONTINUOUS PROJECTIONS ONTO IDEALCONVERGENT SEQUENCES
PAOLO LEONETTI
Abstract.
Let
I ⊆ P ( ω ) be a meager ideal. Then there are no continuousprojections from ℓ ∞ onto the set of bounded sequences which are I -convergentto . In particular, it follows that the set of bounded sequences statisticallyconvergent to is not isomorphic to ℓ ∞ . Introduction
A closed subspace X of a Banach space B is said to be complemented in B if there exists a continuous projection from B onto X . It is known that c , thespace of real sequences convergent to , is not complemented in ℓ ∞ , cf. [10, 12].The aim of this note is to show the ideal analogue of this result.Let I ⊆ P ( ω ) be an ideal, that is, a family closed under subsets and finiteunions. It is also assumed that Fin := [ ω ] <ω ⊆ I and ω / ∈ I . Set I + := P ( ω ) \ I .In particular, each I can be regarded as a subset of the Cantor space ω with theproduct topology, so we can speak of Borel ideals, F σ ideals, etc. An ideal I issaid to be a P-ideal if it is σ -directed modulo finite sets, i.e., for each sequence ( A n ) in I there exists A ∈ I such that A n \ A is finite for all n ∈ ω . We refer to[7] for a recent survey on ideals and filters.A real sequence ( x n ) is said to be I -convergent to y if { n : x n / ∈ U } ∈ I for allneighborhoods U of y . We denote by c ( I ) [resp. c ( I ) ] the space of real sequenceswhich are I -convergent [resp. I -convergent to ]. The set of bounded real I -convergent sequences has been studied, e.g., in [2, 6, 8]. By an easy modificationof [8, Theorem 2.3], c ( I ) ∩ ℓ ∞ is a closed linear subspace of ℓ ∞ (with the supnorm).The question addressed here, posed at the open problem session of the 45thWinter School in Abstract Analysis (Czech Republic, 2017), follows: Question 1. Is c ( I ) ∩ ℓ ∞ complemented in ℓ ∞ ?Before proving our main result, we recall the following: Mathematics Subject Classification.
Primary: 40A35, 46B03. Secondary: 54A20,46B26.
Key words and phrases.
Meager ideal, I -maximal almost disjoint family, complementability,asymptotic density zero sets, I -convergent sequence. Paolo Leonetti
Lemma 1.1.
An infinite dimensional subspace X of ℓ ∞ is complemented in ℓ ∞ if and only if it is isomorphic to ℓ ∞ .Proof. See [1, Proposition 2.5.2 and Theorem 5.6.5]. (cid:3)
Hence, Question 1 can be reformulated as:
Question 2. Is c ( I ) ∩ ℓ ∞ isomorphic to ℓ ∞ ?We will prove that the answer is negative for a large class of ideals. To state ourresult, we recall that a family A ⊆ I + is said to be I -maximal-almost-disjoint (in short, I - mad ) if A is a maximal family (with respect to inclusion) such that A ∩ B ∈ I for all distinct A, B ∈ A , so that for each X ∈ I + there exists A ∈ A such that X ∩ A ∈ I + . (The minimal cardinality a ( I ) of an I - mad hasbeen studied in the literature: e.g., it is known that, if I is an analytic P-ideal, a ( I ) > ω if and only if I is F σ , cf. [4, 5].)Our main result follows: Theorem 1.2.
Let I be an ideal for which there exists an uncountable I - mad family. Then c ( I ) ∩ ℓ ∞ is not complemented in ℓ ∞ . It can be shown that, if I is a meager ideal, there is an I - mad family ofcardinality c , see Lemma 2.3 below. In particular Corollary 1.3. c ( I ) ∩ ℓ ∞ is not complemented in ( and not isomorphic to ) ℓ ∞ whenever I is meager. As an important example, the family of asymptotic density zero sets Z := { S ⊆ ω : | S ∩ [1 , n ] | /n → } is an analytic P-ideal, hence meager. Therefore: Corollary 1.4.
The set of bounded real sequences statistically convergent to ( i.e., c ( Z ) ) is not is isomorphic to ℓ ∞ . Lastly, we obtain an analogue of the main result in [9] (for summability matri-ces):
Corollary 1.5. c is complemented in c ( I ) ∩ ℓ ∞ if and only if I = Fin . It is worth noting that Theorem 1.2 cannot be extended to all ideals I . Indeed, if I is maximal, then the set of bounded I -convergent sequences, which is isomorphicto c ( I ) ∩ ℓ ∞ , is exactly ℓ ∞ .2. Preliminaries and Proofs
Thanks to Lemma 1.1, a negative question to Question 1 would follow if c ( I ) ∩ ℓ ∞ was separable (indeed ℓ ∞ is nonseparable, hence they cannot be isomorphic).However, this works only if I = Fin : Lemma 2.1. c ( I ) is separable if and only if I = Fin . rojections onto ideal convergent sequences Proof.
The if part is known. Conversely, let us suppose that there exists A ∈I ∩ [ ω ] ω . For each X ⊆ ω and ε > , let B ( X , ε ) be the open ball with center X and radious ε . The collection B := { B ( X , /
2) : X ∈ [ A ] ω } is an uncountablefamily of nonempty open sets which are pairwise disjoint, hence c ( I ) is notseparable. (cid:3) At this point, recall the following characterization, see [11] and [3, Theorem4.1.2]:
Lemma 2.2. I is a meager ideal if and only if there exists a finite-to-one function f : ω → ω such that f − ( A ) ∈ I if and only if A is finite. In other words, the second condition is
Fin ≤ RB I , where ≤ RB is the Rudin–Blass ordering. This is sufficient to prove the existence of an uncountable I - mad family: Lemma 2.3.
There exists an I - mad family of cardinality c , provided I is meager.Proof. It is known that there is a
Fin - mad family A of cardinality c , cf. [12].Then, thanks to Lemma 2.2, there exists a finite-to-one function f : ω → ω suchthat f − ( A ) ∈ I if and only if A is finite, hence { f − ( A ) : A ∈ A } is the claimed I - mad family. (cid:3) Let us prove our main result:
Proof of Theorem 1.2.
Let us suppose for the sake of contradiction that c ( I ) ∩ ℓ ∞ is complemented in ℓ ∞ and denote by π : ℓ ∞ → c ( I ) ∩ ℓ ∞ the canonical projection. Define T := I − π , hence T is bounded linear operatorsuch that T ( x ) = 0 for each x ∈ c ( I ) ∩ ℓ ∞ . Note also that, if B / ∈ I , then B is a bounded sequence which is not I -convergent to , hence π ( B ) = B and T ( B ) = 0 .At this point, let ( A j : j ∈ J ) be an uncountable I - mad family, which exists byhypothesis. We are going to show that there exists j ∈ J such that T ( A j ) = 0 ,which is impossible since A j ∈ I + . Indeed, let us suppose that, for each j ∈ J ,there exists x j = ( x j,n ) ∈ ℓ ∞ supported on A j with T ( x j ) = 0 and, withoutloss of generality, k x j k ∞ = 1 . It follows that there exists m, k ∈ ω such that ˜ J := { j ∈ J : | x j,m | ≥ − k } is uncountable. Also, by possibly replacing x j with − x j , let us suppose without loss of generality that x j,m > for all j ∈ ˜ J .For each nonempty finite set F ⊆ ˜ J , define s F = ( s F,n ) := P j ∈ F x j . In partic-ular, k T ( s F ) k ∞ ≥ s F,m ≥ | F | − k . (1)Note also that I := S ( A i ∩ A j ) , where the sum is extended over all distinct i, j ∈ F , belongs to I . This implies that the sequence s F ↾ I is I -convergent to , Paolo Leonetti hence T ( s F ) = T ( s F ↾ I c ) . Therefore k T ( s F ) k ∞ = k T ( s F ↾ I c ) k ∞ ≤ k T k · k s F ↾ I c k ∞ ≤ k T k , which, together with (1), implies | F | ≤ k k T k . This contradicts the fact the ˜ J isinfinite. (cid:3) Proof of Corollary 1.5.
There is nothing to prove if I = Fin . Conversely, fix I ∈ I \ Fin and define X := { x ∈ ℓ ∞ : x i = 0 only if i ∈ I } and Y := X ∩ c . Itis clear that c ⊆ Y ⊆ X ⊆ c ( I ) ∩ ℓ ∞ and that X and Y are isometric to ℓ ∞ and c , respectively. Hence, it is knownthat c can be projected continuously onto Y , let us say through T , see [10].To conclude the proof, let us suppose that there exists a continuous projection H : c ( I ) ∩ ℓ ∞ → c . Then the restriction T ◦ H ↾ X is a continuous projection ℓ ∞ → c . This contradicts Theorem 1.2 (in the case I = Fin ). (cid:3) Acknowledgments.
The author is grateful to Tommaso Russo (Universitàdegli Studi di Milano, IT) for suggesting Question 1 and Lemma 1.1.
References
1. F. Albiac and N. J. Kalton,
Topics in Banach space theory , Graduate Texts in Mathematics,vol. 233, Springer, New York, 2006.2. A. Bartoszewicz, S. Głab, and A. Wachowicz,
Remarks on ideal boundedness, convergenceand variation of sequences , J. Math. Anal. Appl. (2011), no. 2, 431–435.3. T. Bartoszyński and H. Judah,
Set theory , A K Peters, Ltd., Wellesley, MA, 1995, On thestructure of the real line.4. J. E. Baumgartner,
Iterated forcing , Surveys in set theory, London Math. Soc. Lecture NoteSer., vol. 87, Cambridge Univ. Press, Cambridge, 1983, pp. 1–59.5. B. Farkas and L. Soukup,
More on cardinal invariants of analytic P -ideals , Comment. Math.Univ. Carolin. (2009), no. 2, 281–295.6. R. Filipów and J. Tryba, Ideal convergence versus matrix summability , Studia Math., toappear.7. M. Hrušák,
Combinatorics of filters and ideals , Set theory and its applications, Contemp.Math., vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 29–69.8. P. Kostyrko, M. Mačaj, T. Šalát, and M. Sleziak, I -convergence and extremal I -limitpoints , Math. Slovaca (2005), no. 4, 443–464.9. J. Lindenstrauss, Mathematical Notes: A Remark Concerning Projections in SummabilityDomains , Amer. Math. Monthly (1963), no. 9, 977–978.10. A. Sobczyk, Projection of the space ( m ) on its subspace ( c ) , Bull. Amer. Math. Soc. (1941), 938–947.11. M. Talagrand, Compacts de fonctions mesurables et filtres non mesurables , Studia Math. (1980), no. 1, 13–43.12. R. Whitley, Mathematical Notes: Projecting m onto c , Amer. Math. Monthly (1966),no. 3, 285–286. rojections onto ideal convergent sequences Department of Statistics, Università “L. Bocconi”, via Roentgen 1, 20136 Milan,Italy
E-mail address : [email protected] URL ::