HARMONIC-ANALYSIS IN BANACH MODULES .1. GENERAL-PROPERTIES

In a first part, we study for a Banach space the compatibility between the action of a locally compact abelian group and the one of its algebra L(1)(G). Such a space is called a L(1)-module with a compatible G-action. The elements of the module on which G acts continuously play an important role and their characterization is the subject of the second part. We obtain, in particular, a generalization of a result by M. and R. P. BOAS to uniformly continuous fonctions. The last part deals with the Beurling spectrum The space of the elements with a void spectrum admits generally a topological supplement. This result will be very useful for the sudy of the almost-periodic elements and the development on an ''ergodic'' Fourier transform, both subjects being a sequel of this work. We give an application of the spectral theory to problems of automatic continuity, completing B. E. JOHNSON's results.