Jacquet-Langlands and unitarisability

We study a local Jacquet-Langlands correspondence for all smooth irreducible representations. This correspondence is characterized by the fact that it respects the classical Jacquet-Langlands correspondence and it commutes with the parabolic induction functor. It has good behavior with respect to the Jacquet's functor and the involution of Aubert-Schneider-Stuhler. Using this correspondence, we prove some particular cases of the global Jacquet-Langlands correspondence and we deduce that a certain class of representations of an inner form of GL(n) over a p-adic field are unitarizable. This is the first step towards the proof of Conjecture Ul of Tadic.