Power boundedness in Fourier and Fourier–Stieltjes algebras and other commutative Banach algebras

We study power boundedness in the Fourier and Fourier–Stieltjes algebras, A(G) and B(G), of a locally compact group G as well as in some other commutative Banach algebras. The main results concern the question of when all elements with spectral radius at most one in any of these algebras are power bounded, the characterization of power bounded elements in A(G) and B(G) and also the structure of the Gelfand transform of a single power bounded element.