In the first part of this work, we give some criteria of automatic continuity for representations from topological groups in Banach algebras. Two different approaches are used : the first one, based on the Glicksberg-De Leeuw decomposition, applies to locally compact groups; the second one, based using an equicontinuity result for sequences of positive definite functions applies to Polish (perhaps non locally compact) groups. Typically, the continuity of a representation is expressed through the continuity of the composition of this representation with some functionals on the representation algebra. Some results for group morphisms are deduced. In the second part, the results of the first part are applied to obtain properties of the spectra of the elements in the range of the representation outside a "small" (in various sense) subset of the group in the abelian case. The third section of this work partially generalizes the results of the second part to Lie groups (non abelian in general) refining a theorem obtained by J.M. Paoli and J.C. Tomasi in a previous work. Keywords: locally compact groups, Polish groups, Lie groups, Banach algebras, group representation, automatic continuity, Spectrum of operators.
