Power boundedness in Banach algebras associated with locally compact groups
Let $G$ be a locally compact group and $B(G)$ the Fourier–Stieltjes algebra of $G$. Pursuing our investigations of power bounded elements in $B(G)$, we study the extension property for power bounded elements and discuss the structure of closed sets in the coset ring of $G$ which appear as $1$-sets of power bounded elements. We also show that $L^1$-algebras of noncompact motion groups and of noncompact IN-groups with polynomial growth do not share the so-called power boundedness property. Finally, we give a characterization of power bounded elements in the reduced Fourier–Stieltjes algebra of a locally compact group containing an open subgroup which is amenable as a discrete group.