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Subprojective Banach Spaces Subprojective Banach spaces T. Oikhberg Dept. of Mathematics, University of Illinois at Urbana-Champaign, Urbana IL 61801, USA E. Spinu Dept. of Mathematical and Statistical Sciences, University of Alberta Edmonton, Alberta T6G 2G1, CANADA Abstract A Banach space X is called subprojective if any of its infinite dimensional sub- spaces contains a further infinite dimensional subspace complemented in X. This paper is devoted to systematic study of subprojectivity. We examine the stability of subprojectivity of Banach spaces under various operations, such us direct or twisted sums, tensor products, and forming spaces of operators. Along the way, we obtain new classes of subprojective spaces. Keywords: Banach space, complemented subspace, tensor product, space of operators. 1. Introduction and main results Throughout this note, all Banach spaces are assumed to be infinite dimen- sional, and subspaces, infinite dimensional and closed, until specified otherwise. A Banach space X is called subprojective if every subspace Y X contains a further subspace Z Y , complemented in X. This notion was introduced in [41], in order to study the (pre)adjoints of strictly singular operators. Recall that an operator T P BpX; Y q is strictly singular (T P SSpX; Y q) if T is not an isomorphism on any subspace of X. In particular, it was shown that, if Y is subprojective, and, for T P BpX; Y q, T ¦ P SSpY ¦;X¦q, then T P SSpX; Y q. Later, connections between subprojectivity and perturbation classes were discovered. More specifically, denote by Φ pX; Y q the set of upper semi-Fredholm operators { that is, operators with closed range, and finite dimensional kernel. If Φ pX; Y q H, we define the perturbation class P Φ pX; Y q tS P BpX; Y q : T S P Φ pX; Y q whenever T P Φ pX; Y qu: Email addresses: [email protected] (T. Oikhberg), [email protected] (E. Spinu) Preprint submitted to Journal of Mathematical Analysis and Applications July 24, 2014 It is known that SSpX; Y q P Φ pX; Y q. In general, this inclusion is proper. However, we get SSpX; Y q P Φ pX; Y q if Y is subprojective (see [1, Theorem 7.51] for this, and for similar connections to inessential operators). Several classes of subprojective spaces are described in [16]. For instance, the spaces `p (1 ¤ p 8) and c0 are subprojective. Common examples of non- subprojective space are L1p0; 1q (since all Hilbertian subspaces of L1 are not complemented), Cp∆q, where ∆ is the Cantor set, or `8 (for the same reason). The disc algebra is not subprojective, since by [43, III.E.3] it contains a copy of Cp∆q. By [41], Lpp0; 1q is subprojective if and only if 2 ¤ p 8. Consequently, the Hardy space Hp on the disc is subprojective for exactly the same values of p. Indeed, H8 contains the disc algebra. For 1 p 8, Hp is isomorphic to Lp. The space H1 contains isomorphic copies of Lp for 1 p ¤ 2 [42, Section 3]. On the other hand, VMO is subprojective ([30], see also [37] for non-commutative generalizations). In this paper we examine several aspects of subprojectivity. We start by collecting various facts needed to study subprojectivity (Section2). Along the way, we prove that subprojectivity is stable under suitable direct sums (Proposi- tion 2.2). However, subprojectivity is not a 3-space property (Proposition 2.8). Consequently, subprojectivity is not stable under the gap metric (Proposition 2.9). Considering the place of subprojective spaces in Gowers dichotomy, we observe that each subprojective space has a subspace with an unconditional basis. However, we exhibit a space with an unconditional basis, but with no subprojective subspaces (Proposition 2.11). In Section3, we investigate the subprojectivity of tensor products, and of spaces of operators. A general result on tensor products (Theorem 3.1) yields q p the subprojectivity on `pb`q and `pb`q for 1 ¤ p; q 8 (Corollary 3.3), as well as of KpLp;Lqq for 1 p ¤ 2 ¤ q 8 (Corollary 3.4). We also prove that the space BpXq is never subprojective (Theorem 3.10), and give an example of r non-subprojective tensor product `2b`2 (Proposition 3.9). Throughout Section4, we work with CpKq spaces, with K compact metriz- able. We begin by observing that CpKq is subprojective if and only if K is scattered. Then we prove that CpK; Xq is subprojective if and only if both CpKq and X are (Theorem 4.1). Turning to spaces of operators, we show that, for K scattered, ΠqppCpKq; `qq is subprojective (Proposition 4.4). We also study continuous fields on a scattered base space, proving that any scattered separable CCR C¦-algebra is subprojective (Corollary 4.7). Section5 shows that, in many cases, subprojectivity passes from a sequence space to the associated Schatten space (Proposition 5.1). Proceeding to Banach lattices, in Section6 we prove that p-disjointly ho- mogeneous p-convex lattices (2 ¤ p 8) are subprojective (Proposition 6.2). In Section7 (Proposition 7.1), we show that the lattice Xp`pq is subprojective whenever X is. Consequently (Proposition 7.3), if X is a subprojective space with an unconditional basis and non-trivial cotype, then RadpXq is subprojec- tive. Throughout the paper, we use the standard Banach space results and nota- 2 tion. By BpX; Y q and KpX; Y q we denote the sets of linear bounded and com- pact operators, respectively, acting between Banach spaces X and Y . UBpXq refers to the closed unit ball of X. For p P r1; 8s, we denote by p1 the \adjoint" of p (that is, 1{p 1{p1 1). 2. General facts about subprojectivity We start collecting general facts about subprojectivity by observing that subprojectivity passes to subspaces. Proposition 2.1. Any subspace of a subprojective Banach space is subprojec- tive. In contrast to this, subprojectivity does not pass to quotients. Indeed, any separable Banach space is a quotient of `1, which is subprojective. As noted in Section1, there exist separable non-subprojective spaces. However, subprojectivity does pass to direct sums. Proposition 2.2. (a) Suppose X and Y are Banach spaces. Then the following are equivalent: 1. Both X and Y are subprojective. 2. X ` Y is subprojective. (b) Suppose X1;X2;::: are Banach spaces, and E is a space with a 1-uncon- ditional basis. Then the following are equivalent: 1. The spaces E;X1;X2;::: are subprojective. ° p q 2. n Xn E is subprojective. }¤} ° In (b), we view E as a space of all sequences, for which± the norm E is finite. p XnqE refers to the space of all sequences pxnqnP P P Xn, endowed with n N n N }p q P } }p} } q} the norm xn n N xn Xn E . Due° to the 1-unconditionality (actually, 1- p q suppression unconditionality suffices), n Xn E is a Banach space. In the proof of Proposition 2.2, and further in the paper, we will use two results stated below. Proposition 2.3. Consider Banach spaces X and X1, and T P BpX; X1q. Sup- pose Y is a subspace of X, T |Y is an isomorphism, and T pY q is complemented in X1. Then Y is complemented in X. Proof. If Q is a projection from X1 to T pY q, then T ¡1QT is a projection from X onto Y . This immediately yields: 3 Corollary 2.4. Suppose X and X1 are Banach spaces, and X1 is subprojective. Suppose, furthermore, that Y is a subspace of X, and there exists T P BpX; X1q so that T |Y is an isomorphism. Then Y contains a subspace complemented in X. The following version of \Principle of Small Perturbations" is folklore, and essentially contained in [5]. We include the proof for the sake of completeness. Proposition 2.5. Suppose pxkq is a seminormalized basic sequence in a Banach space X, and pykq is a sequence so that limk }xk¡yk} 0. Suppose, furthermore, r P s that every subspace of span yk : k N contains a subspace complemented in X. r P s Then span xk : k N contains a subspace complemented in X. Proof. Replacing xk by xk{}xk}, we can assume that pxkq is normalized. De- ¦ } ¦} note the biorthogonal functionals° by xk , and set K supk xk . Passing to a } ¡ } {p q subsequence, we can assume° that k xk yk 1 2K . Define the operator P p q ¦p qp ¡ q } } { U B X by setting Ux k xk x yk xk . Clearly U 1 2, and therefore, V IX U is invertible. Furthermore, V xk yk. If Q is a projection from X r P s ¡1 onto a subspace W span yk : k N , then P V QV is a projection from r P s X onto a subspace Z span xk : k N . Remark 2.6. Note that, in the proof above, the kernels and the ranges of the projections Q and P are isomorphic, via the action of V . Proof of Proposition 2.2. Due to Proposition 2.1, in both (a) and (b), only the implication p2q ñ p1q needs to be established. (a) Throughout the proof, PX and PY stand for the coordinate projections from X ` Y onto X and Y , respectively. We have to show that any subspace E of X ` Y contains a further subspace G, complemented in X ` Y . Show first that E contains a subspace F so that either PX |F or PY |F is an isomorphism. Indeed, suppose PX |F is not an isomorphism, for any such F .
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