Abstract
We introduce two new local ℓ1-indices of the same type as the Bourgain ℓ1-index; the ℓ+1-index and the ℓ+1-weakly null index. We show that the ℓ+1-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain ℓ1. The ℓ+1-weakly null index has the same form as the Bourgain ℓ1-index: if it is countable it must take values ωα for some α<ω1. The different ℓ1-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its ℓ1-index, via the ℓ+1-weakly null index. We show that I(C(ωωα))=ω^1+α+1.
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Alspach, D., Judd, R. & Odell, E. The Szlenk index and local ℓ1-indices. Positivity 9, 1–44 (2005). https://doi.org/10.1007/s11117-002-9781-0
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DOI: https://doi.org/10.1007/s11117-002-9781-0