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After the papers [2] and [-3] had gone to press, the authors generalized these results to Brownian paths in separable Banach spaces and strengthened the limiting process to almost sure convergence. Through a communication with A. de Acosta we have learned that the latter result follows by an easy modification of the techniques used to prove Theorem 9 of [-1], which gives priority for this result to the authors of [,1]. Our proof (with V. Goodman) of the Banach space result is of some interest. The outer law, together with the one-dimensional inner law for the line passing through an arbitrary point on the surface of the unit ball of the kernel space, are used to deduce the full inner law by a simple compactness and scaling argument.

Abstract. For an element x in a Banach algebra we study the condition (1) sup |jr"|| < oo.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 8D. There follows a localization theorem for commutative, regular, semisimple algebras A which says that whether or not (1) is satisfied for an element x e A with spectral radius 1 is determined by the behavior of its Gelfand transform x on any neighborhood of the points where |jc| = 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\G) satisfying certain differentiability conditions. We conclude with the result that, for a certain class 3? of locally compact groups containing all abelian and all compact groups, a group G e 'S has the property that every function in L^tfi) with spectral radius one satisfies (1) if and only if G is compact and abelian.

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