On the Algebraic Structures in A_Φ (G)

Let G be a locally compact group and (Φ , Ψ ) be a complementary pair of N -functions. In this paper, using the powerful tool of porosity, it is proved that when G is an amenable group, then the Figà–Talamanca–Herz–Orlicz algebra A_Φ(G) is a Banach algebra under the convolution product if and only if G is compact. Then, it is shown that A_Φ(G) is a Segal algebra, and as a consequence, the amenability of A_Φ(G) and the existence of a bounded approximate identity for A_Φ(G) under the convolution product is discussed. Furthermore, it is shown that for a compact abelian group G , the character space of A_Φ(G) under the convolution product can be identified with G , the dual of G .