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ℓ ∞ -sums and the Banach space ℓ ∞ / c ₀

Christina Brech, Piotr Koszmider (2014)

Fundamenta Mathematicae

This paper is concerned with the isomorphic structure of the Banach space ℓ ∞ / c ₀ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ℓ ∞ / c ₀ does not have an orthogonal ℓ ∞ -decomposition, that is, it is not of the form ℓ ∞ ( X ) for any Banach space X. The main local result is that it is consistent that ℓ ∞ ( c ₀ ( ) ) does not embed isomorphically into ℓ ∞ / c ₀ , where is the cardinality of the continuum, while ℓ ∞ ...

ℓ¹-Spreading models in subspaces of mixed Tsirelson spaces

Denny H. Leung, Wee-Kee Tang (2006)

Studia Mathematica

We investigate the existence of higher order ℓ¹-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space X = T [ ( θ ₙ , ₙ ) n = 1 ∞ ] : (1) Every block subspace of X contains an ℓ ¹ - ω -spreading model, (2) The Bourgain ℓ¹-index I b ( Y ) = I ( Y ) > ω ω for any block subspace Y of X, (3) l i m ₘ l i m s u p ₙ θ m + n / θ ₙ > 0 and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X. Moreover, if one (and hence all) of these conditions...

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