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A uniformly convex hereditarily indecomposable Banach space

A {\em hereditarily indecomposable (or H.I.)} Banach space is an infinite dimensional Banach space such that no subspace can be written as the topological sum of two infinite dimensional subspaces. As an easy consequence, no such space can contain an unconditional basic sequence. This notion was first introduced in 1993 by T.Gowers and B.Maurey, who constructed the first known example of a hereditarily indecomposable space. Gowers-Maurey space is reflexive, however it is not uniformly convex. In this article, we provide an example of a uniformly convex hereditarily indecomposable space, using similar methods as Gowers and Maurey, and using the theory of complex interpolation for a family of Banach spaces of Coifman, Cwikel, Rochberg, Sagher and Weiss.