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We strengthen and improve theorems of Choquet-Deny and of Foguel concerning convolution equations and iterates of convolution of a measure to all algebras      A p ( G ) {A_p}(G)       and all locally compact groups. Furthermore, we improve results of H. P. Rosenthal on ideals of      A ( G ) A(G)       to the algebras      A p ( G ) {A_p}(G)       and show that some hold for amenable groups but not for free nonabelian groups. Finally, we improve a (possibly) weak version of a theorem of Gilbert on projections onto some subspaces of      L ∞  ( G ) {L^\infty }(G)       to all locally compact groups.

On some properties of the Banach algebras 𝐴_{𝑝}(𝐺) for locally compact groups