Henry Helson

Isomorphisms of abelian group algebras

Let G and H be locally compact groups with group algebras L (G)and L (H) respectively. If G and H are isomorphic groups, the correspondence between points of G and H given by the isomorphism induces an isomorphism of the group algebras. The purpose of this paper is to make a contribution to the converse question: assuming that L (G) and L ( H ) a r e isomorphic algebras, under what conditions can it be asserted that the underlying groups are isomorphic ? As a case of the problem, one can ask when an automorphism of a single group algebra is induced by an automorphism of the underlying group. A discussion appeared in the author s thesis (Harvard, 1950) 1 , where the group was assumed to be commutative. The principal result there stated was Theorem 3 of this paper, except that only automorphisms were considered. Further information is hard to obtain even for simple groups; for example, i t is not known whether there are any automorphisms of the algebra on the line, except a few obvious and trivial ones. At a late stage of this work I learned that J. WENDEL has obtained Theorem 3 (for an operator assumed to be isometric) even for non-abelian group algebras. More recently he has established all of Theorem 3. While his methods and mine undoubtedly are related, it is difficult to make direct comparisons because of the complexity of the general case. I am grateful to Dr. WENDEL for correspondence about the problem, and for a summary of [7] before it appeared in print. We shall consider only abelian groups, where the Fourier transform is a convenient tool. Our m a i n result, Theorem 4, asserts the following: If T is an operator mapping L (G) isomorphically onto L (H) with bou~ul less than two, where G and H are locally compact abelian groups, and if the dual group of G or of H is connected, then G and H are isomorphic groups, and T is the natural isomorphism of algebras induced by the group isomorphism. Since Theorem 3 is a corol lary of the more complicated methods used here, we are not giving its original proof. The argument used to prove Theorem 4 is a modification of a proof of A. BEURLI~rG for the following theorem: if for each real 2

Foundations of the theory of Dirichlet series

1. A modern reader, familiar with the methods of functional analysis, is struck with the conviction tha t the classical theory of Dirichlet series [3, 5, 6] must have content expressible in more congenial language. Harald Bohr recognized the analogy between harmonic series and the Fourier series of functions on the circle; later his theory of almostperiodic functions was shown to be par t of a theory of Fourier series on compact abelian groups tha t embraced the classical case of the circle group as well. Various generalizations t reat the spaces L ~, and it is fairly clear by now how much of Fourier series can be developed in the more general setting. Nevertheless another par t of the theory of Dirichlet series, to which Bohr himself contributed a great deal, does not seem to be harmonic analysis. This is the par t depending on the Dirichlet condition

Convergent Dirichlet series

which are loosely connected by a common idea of proof, rather than by any formal dependence. Section 2 contains an elementary lemma, really a reformulation of the well-known formula expressing the abscissa of convergence of (1) in terms of the coefficients and exponents of the series. After this lemma, the sections can be read independently. Section 3 treats the convergence problem for Dirichlet series: to determine the abscissa of convergence from properties of the function ]. A famous and difficult theorem of Landau and Schnee gives su//icient conditions for (1) to converge in a region a > a 0. A restriction has to be put on the exponents 2n as well as on ]. Without any restriction on the exponents beyond the fundamental one

Analyticity on Flows

Publisher Summary This chapter discusses the analyticity on flows. The chapter defines analyticity on a flow in two ways, and proves the existence of analytic functions by means of the spectral theorem. A simple proof that the bounded analytic functions form a weak-star Dirichlet algebra is given. Almost all the theory of Hardy spaces and cocycles can be restated for ergodic flows. There is one problem that although the cocycles lead to invariant subspaces, new information is needed to show that every invariant subspace is so obtained. The chapter uses the existence of smooth analytic functions to prove part of the theorem of Ambrose on flows built under a function.