We construct a reflexive Banach space XD with an unconditional basis such that all spreading models admitted by normalized block sequences in XD are uniformly equivalent to the unit vector basis of ₁, yet every infinite-dimensional closed subspace of XD fails the Lebesgue property. This is a new result in a program initiated by Odell in 2002 concerning the strong separation of asymptotic properties in Banach spaces.
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Gaebler et al. (2024) studied this question.
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