The Rajchman algebra B0(G) of a locally compact group G

Let G   be a locally compact group, B(G)B(G) the Fourier–Stieltjes algebra of G   and B0(G)=B(G)∩C0(G)B0(G)=B(G)∩C0(G). The space B0(G)B0(G) is a closed ideal of B(G)B(G). In this paper, we study the Banach algebra B0(G)B0(G) under various aspects. The main emphasis is on regularity and the existence of various kinds of approximate identities, the question of when the quotient of B0(G)B0(G) modulo the Fourier algebra A(G)A(G) is radical, the Bochner–Schoenberg–Eberlein property and a characterization of elements in B0(G)B0(G) in terms of continuity of translation properties. The paper also contains a number of illustrating examples.