Banach spaces in various positions

Abstract

We formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index a(Y, X) that measures how many non-equivalent positions Y admits in X, and obtain estimates of a(Y, X) for X a classical Banach space such as p , L p , L 1 , C(ω ω ) or C[0, 1]. Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from c 0 or 2 ? Recall that a Banach space X is said to be automorphic if every subspace Y admits only one position in X; i.e., a(Y, X) = 1 for every subspace Y of X. We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an L ∞ -space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both X and X * are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either c 0 or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either L ∞ or a superreflexive type 2 Banach lattice.

Characterizations of Strictly Singular and Strictly Cosingular Operators by Perturbation Classes

“…REMARK 3.4. We observe that the coincidence of SS(X, Y ) and P + (X, Y ) in some cases is related with the different positions in which a Banach space can be embedded as a subspace of another Banach space, as studied in [6].…”

On the classification of positions and complex structures in Banach spaces

“…The papers [5,18,6,4] were devoted to the study of different aspects of the automorphic problem. In [18,6] in particular, it is provided a general theory of positions for subspaces of a Banach space, by defining equivalent embeddings. Namely, given two infinite codimensional embeddings T, U : Y → X between separable Banach spaces, we let ∼ be the equivalence relation: T ∼ U if and only if there exists an automorphism A of X such that T = AU.…”

“…The automorphy index of X is defined as a(X) = sup Y a(Y, X) and, of course, a Banach space is said to be automorphic if a(X) = 1. In [6] it is estimated the automorphy indices a(Y, X) for classical Banach spaces. The authors obtain, among other results: a(c 0 , X) = {0, 1, 2, ℵ 0 } for every separable Banach space X; a(Y, ℓ p ) = c for all subspaces of ℓ p , p = 2, and a(Y, L p ) = c for all subspaces of L p , p > 2 not isomorphic to ℓ 2 ; while a(ℓ 2 , L p ) = 1; for 1 < p < 2 one has a(Y, L p ) = c for all nonstrongly embedded subspaces of L p ; a(Y, L 1 ) = c for all nonreflexive subspaces of L 1 , while a(ℓ 2 , L 1 ) = c; a(Y, C[0, 1]) ∈ {1, c} for every separable Banach space Y .…”

“…We shall need the notion of uniformly finitely extensible space considered in [6]. A space is uniformly finitely extensible (or UFO) if there exists λ ≥ 1 such that for every finite dimensional subspace E ⊂ X, each linear operator t : E → X may be extended to a linear operator T : X → X with T ≤ λ t .…”

$\aleph$-injective Banach spaces and $\aleph$-projective compacta

“…If we deal with ℵ-injectivity instead of (1, ℵ)-injectivity, the matter becomes more complicated: since C(N * ) contains an uncomplemented copy of itself [10] it is not c + -injective. We do not know whether it is consistent that C(N * ) is ℵ 2 -injective.…”