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The goal of this paper is to study Banach algebra valued cocycles
over
Anosov actions.
The study of cohomological equations over Anosov diffeomorphisms and flows
was started in two influential papers by Livsic ([L1], [L2]), and
experienced later a tremendous development. This paper is a continuation
of
[NT1] and [NT2]. We show here that the techniques used to study cocycles
with
values in Lie groups, and with values in diffeomorphism groups, can be
adapted to Banach algebra valued cocycles. Results of this nature are
necessary for the study of extensions of Anosov group actions on infinite
dimensional manifolds.