Measures with Bounded Convolution Powers

Abstract

For an element x in a Banach algebra we study the condition \begin{equation}\tag {$1$} \sup \limits _{n \geqq 1} \left \| {{x^n}} \right \| < \infty .\end{equation} Although our main results are obtained for the algebras $M(G)$ of finite complex measures on a locally compact abelian group, we begin by considering the question of bounded powers from the point of view of general Banach-algebra theory. We collect some results relating to (1) for an element whose spectrum lies in the unit disc D and has only isolated points on $\partial D$. There follows a localization theorem for commutative, regular, semisimple algebras A which says that whether or not (1) is satisfied for an element $x \in A$ with spectral radius 1 is determined by the behavior of its Gelfand transform $\hat x$ on any neighborhood of the points where $|\hat x| = 1$. We conclude the general theory with remarks on the growth rates of powers of elements not satisfying (1). After some applications of earlier results to the algebras $M(G)$, we prove our main theorem. Namely, we obtain strong necessary conditions on the Fourier transform for a measure to satisfy (1). Some consequences of this theorem and related results follow. Via the generalization of a result of G. Strang, sufficient conditions for (1) to hold are obtained for functions in ${L^1}(G)$ satisfying certain differentiability conditions. We conclude with the result that, for a certain class $\mathcal {G}$ of locally compact groups containing all abelian and all compact groups, a group $G \in \mathcal {G}$ has the property that every function in ${L^1}(G)$ with spectral radius one satisfies (1) if and only if G is compact and abelian.