We construct a hereditarily indecomposable Banach space with dual space isomorphic to ℓ1. Every bounded linear operator on this space is expressible as λI + K with λ a scalar and K compact.

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Abstract. Two examples of asymptotic 1 Banach spaces are given. The first, Xu, has an unconditional basis and is arbitrarily distortable. The second, X, does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson's.

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