The scalar-plus-compact property in spaces without reflexive subspaces

Abstract

Abstract. A hereditarily indecomposable Banach space Xnr is constructed that is the first known example of a L∞-space not containing c0, ℓ1, or reflexive subspaces and answers a question posed by J. Bourgain. Moreover, the space Xnr satisfies the "scalar-plus-compact" property and it is the first known space without reflexive subspaces having this property. It is constructed using the Bourgain-Delbaen method in combination with a recent version of saturation under constraints in a mixed-Tsirelson setting. As a result, the space Xnr has a shrinking finite dimensional decomposition and does not contain a boundedly complete sequence.

An unconditionally saturated Banach space with the scalar-plus-compact property

“…Notice that all the results from Section 3 are valid also for the basis (d γ ) γ∈Γ of the space X Kus , as d γ = Rd γ , R * e * γ = e * γ , γ ∈ Γ by Remark 1.11 [5] and R = 1. By Proposition 1.13 [5] we can use the analysis of nodes introduced in Section 2 in the space X Kus . We write now the precise form of each e * γ depending on the type of the node γ ∈ Γ.…”

“…The space X = X (Γq ,iq)q is defined as X = d γ : γ ∈ Γ , where d γ is given by d γ = i q (e γ ), with q chosen so that γ ∈ Γ q \ Γ q−1 . An efficient method of defining particular examples of BD-L ∞ -spaces as quotients of canonical BD-L ∞ -spaces was given in [5]. The authors proved that given a BD-L ∞ -space X ⊂ ℓ ∞ (Γ) any so-called self-determined set Γ ′ ⊂ Γ produces a further L ∞ -space Y = d γ : γ ∈ Γ \ Γ ′ and a BD-L ∞ -space X/Y , with the quotient map defined by the restriction of Γ to Γ ′ .…”

Symmetrically separated sequences in the unit sphere of a Banach space

“…with b a unit block, or there exists a symmetrically (1 + ε 2 )-separated finite family of unit blocks b 1 < b 2 < · · · < b n that admits no such extension. In the first case, the proof is completed by the simple induction argument, while in the second one we have obtained a family B 2 := (b (2)…”

“…More recently, Argyros and Motakis ( [2]) constructed a L ∞ -space X AM without reflexive subspaces whose dual is isomorphic to ℓ 1 . In particular, X AM is an Asplund space containing weakly Cauchy sequences that do not converge weakly, so the unit ball of X AM is not completely metrisable in the relative weak topology.…”

Relações entre subespaços, ciclicidade e hiperciclicidade em espaços de Banach

“…Demonstração: Imediato das duas últimas proposições. Outros exemplos de espaços com pouquíssimos operadores são X nr e X Kus construídos, respectivamente, no preprint [4] e no artigo [19]. Demonstração:…”