This article by Pierre Eymard, published in the *Bulletin de la S. M. F.* in 1964, extends harmonic analysis results from abelian groups to general locally compact groups. The main focus is on the Fourier algebra \( B(G) \) and the Fourier-Stieltjes algebra \( A(G) \) of a locally compact group \( G \). Key points include: - The Fourier algebra \( B(G) \) is defined as the algebra of continuous functions of positive type on \( G \), equipped with a norm derived from the \( C^* \)-algebra structure of \( G \). - The Fourier-Stieltjes algebra \( A(G) \) is the subalgebra of \( B(G) \) generated by continuous functions of compact support. - The topological space of maximal ideals of \( A(G) \) is identified with \( G \). - The article discusses properties of \( B(G) \) and \( A(G) \), including the existence of partitions of unity and the identification of \( A(G) \) with the set of functions \( f \star \tilde{g} \) where \( f \) and \( g \) are in \( L^2(G) \). - It also covers the synthesis spectrum, generalizing results from the abelian case to non-abelian groups, and extends the Wiener-Godement theorem and the Beurling-Kaplansky theorem. - The article concludes with an extension of the Ditkin theorem, though it notes that this result holds only for operators satisfying a certain condition. The work is a significant contribution to the theory of harmonic analysis on locally compact groups, providing a comprehensive framework for understanding these algebras and their properties.This article by Pierre Eymard, published in the *Bulletin de la S. M. F.* in 1964, extends harmonic analysis results from abelian groups to general locally compact groups. The main focus is on the Fourier algebra \( B(G) \) and the Fourier-Stieltjes algebra \( A(G) \) of a locally compact group \( G \). Key points include: - The Fourier algebra \( B(G) \) is defined as the algebra of continuous functions of positive type on \( G \), equipped with a norm derived from the \( C^* \)-algebra structure of \( G \). - The Fourier-Stieltjes algebra \( A(G) \) is the subalgebra of \( B(G) \) generated by continuous functions of compact support. - The topological space of maximal ideals of \( A(G) \) is identified with \( G \). - The article discusses properties of \( B(G) \) and \( A(G) \), including the existence of partitions of unity and the identification of \( A(G) \) with the set of functions \( f \star \tilde{g} \) where \( f \) and \( g \) are in \( L^2(G) \). - It also covers the synthesis spectrum, generalizing results from the abelian case to non-abelian groups, and extends the Wiener-Godement theorem and the Beurling-Kaplansky theorem. - The article concludes with an extension of the Ditkin theorem, though it notes that this result holds only for operators satisfying a certain condition. The work is a significant contribution to the theory of harmonic analysis on locally compact groups, providing a comprehensive framework for understanding these algebras and their properties.