  Let E be a linear isometric representation of a group \Gamma. In this paper
we construct and study a family of quasicocycles \Gamma -> E that arise from
splittings \Gamma = A * B. Under certain assumptions on A, B and E the bounded
cohomology classes associated to these quasicocycles form an
infinite-dimensional subspace of H^2_b(\Gamma,E). This is in particular the
case when \Gamma is free and E finite-dimensional or of the type l^p(\Gamma).
For the trivial target E = R we obtain a new family of quasimorphisms for which
we compute the Gromov norm in bounded cohomology. This yields a linear
isometric embedding D(A) \oplus D(B) -> H^2_b(\Gamma,R), where D(A) is a Banach
space which is norm-equivalent to the alternating subspace of l^inf(A). We
prove that there are classes of our type in H^2_b(F_2,R) which have infinite
stabilizer under the natural action of Out(F_2). By replacing the target E with
a group G with bi-invariant metric we obtain a new type of
quasi-representations \Gamma -> G that arise from splittings of \Gamma.
