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

2016/08/05 by Argyros, Spiros A., Motakis, Pavlos
#46B03 #46B06 #46B25 #46B45 (Primary) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1608.01962

Abstract

A hereditarily indecomposable Banach space \mathfrakX_\mathfraknr is constructed that is the first known example of a \mathscrL_∞-space not containing c₀, ℓ₁, or reflexive subspaces and answers a question posed by J. Bourgain. Moreover, the space \mathfrakX_\mathfraknr 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 \mathfrakX_\mathfraknr has a shrinking finite dimensional decomposition and does not contain a boundedly complete sequence.

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