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In this paper we give an explicit description of the universal unitarycompletion of certain locally Q_p-analytic representations of GL_2(F), where Fis a finite extension of Q_p (this generalizes some results of Berger-Breuilfor F=Q_p). To this aim, we make use of certain Banach spaces of C^r functionson O_F (for r a positive real number) introduced by the author.

Sur certains compl\'et\'es unitaires universels explicites pour GL_2(F)