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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Measures with bounded convolution powers

Author: Bertram M. Schreiber
Journal: Trans. Amer. Math. Soc. 151 (1970), 405-431
MSC: Primary 42.50; Secondary 42.55
MathSciNet review: 0264335
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Abstract: For an element x in a Banach algebra we study the condition

$\displaystyle \mathop {\sup }\limits_{n \geqq 1} \left\Vert {{x^n}} \right\Vert < \infty .$ ($ 1$)

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 $ \vert\hat x\vert = 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.

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Keywords: Bounded iterates, convolution powers, singly-generated algebras, spectral projection, semisimple algebra, regular algebra, partition of unity, localization, extreme point, growth rate, Bohr compactification, idempotent measure, coset ring, homomorphism of group algebras, piecewise affine map, compactly generated group, extension of Fourier-Stieltjes transforms, differentiable Fourier transforms, commutator subgroup
Article copyright: © Copyright 1970 American Mathematical Society