we commence by using from a new norm on l1(g) the -algebra of all integrable functions on locally compact group g, to make the c-algebra c(g). consequently, we find its dual b(g), which is a banach algebra so-called fourier-stieltjes algebra, in the set of all continuous functions on g. we consider most of important basic theorems about this algebra. this consideration leads to a rather comprehensive knowledge about the fourier-stieltjes algebra’s ideal, fourier algebra a(g). we study this last algebra and approach to its dual; accordingly, we find von neumann algebra v n(g), the dual of a(g) and consider lots of its important properties. features of v n(g) as a von neumann algebra widen our knowledge not only about v n(g) but also about a(g). studying fourier algebra for locally compact abelian group g, we consider the identify relation between fourier algebra and fourier transformation for locally compact abelian groups. in this section, we also explore relations between a(g) and l1( bg) for abelin group g and its dual locally compact group bg. the mentioned relations bring about similarities between v n(g) and l1(g). as a result, we use from these relations, to settle some subalgebras of v n(g) as uc( bg), w( bg) and ap( bg). in some special cases, we elaborate on these subspaces interrelations. eventually, by defining segal algebras and abstract segal algebras, we establish lebesgue- fourier algebra sa(g). its dramatic property as a segal algebra and even an abstract segal algebra for a(g), is followed in some theorems.