Harmonic analysis in Banach modules .2. Quasi-periodicity and ergodicity

In a first part, we generalize the theorem about approximing the almost periodic functions defined on a locally compact abelian group G by trigonometric polynoms. We use for this purpose the theory of L(1) (G)-modules. We present in addition a generalization of Bohr's almost-periods. In a second part, we define the totally ergodic part of a L(1) (G)-module E. which countains the almost periodic and weakly almost periodic parts studied before. We define on it an ''ergodic Fourier transform''. Calling (G) over bar the Bohr compactification of G, we show that the totally ergodic part of E is a L1 ((G) over bar)-module, whose degenerate part is the kernel of the ergodic Fourier transform. In the case where E = L(infinity) (G), we revisit Eberlein-Jacobs' decomposition of the weakly almost periodic functions: we construct a canonical surjection from the totally ergodic part of L(infinity) (G) upon the corresponding one of L(infinity) ((G) over bar), extending the identification of AP(G) with C((G) over bar).