  The question studied here is the behavior of the Poisson bracket under
C^0-perturbations. In this purpose, we introduce the notion of
pseudo-representation and prove that for a normed Lie algebra, it converges to
a representation. An unexpected consequence of this result is that for many
non-closed symplectic manifolds (including cotangent bundles), the group of
Hamiltonian diffeomorphisms (with no assumptions on supports) has no C^{-1}
bi-invariant metric. Our methods also provide a new proof of Gromov-Eliashberg
Theorem, it is to say that the group of symplectic diffeomorphisms is
C^0-closed in the group of all diffeomorphisms.
