An analytic semigroup version of the Beurling-Helson theorem

Abstract

Introduction The motivation for the work in this paper is a question posed by J. Esterle in [B]. Recall, an analytic semigroup in a Banach algebra B is a B-valued analytic function a defined on the right half-plane H = {z ∈ C : 0} such that a = aa for all z, w ∈ H. The question is the following: If the group algebra (of a non-discrete locally compact group) contains a non-trivial semigroup, analytic on the right half-plane, which is bounded on the line 1 + it, must the group be compact? This question arises from the study of the various semigroups which are to be found in group algebras. More precisely, the type of growth that an analytic semigroup may have on different subsets of H has been discovered to influence the structure of the Banach algebras where the semigroup takes its values, group algebras in particular (see [S] and also [B], [G1], [G2], [G3], [W]). As an example, and related to his classification of radical Banach algebras, Esterle observed that, thanks to the Alfhors-Heins theorem in one complex variable, there cannot exist such algebras containing at the same time semigroups a of polynomial growth on vertical lines in H ([B, pp. 4-65]). This property of the radical algebras had indeed been previously used by Esterle himself in his complex-variable proof of the Wiener tauberian theorem (see [S] for an extended version of it), together with the fact that the group algebra L1(R) has an abundance of such semigroups. For, in this way, L1(R) cannot have any non-trivial radical quotient, whence the Wiener theorem holds. The most important examples of analytic semigroups in L1(R) with polynomial growth on vertical lines are the classical ones. Let us recall that the Gaussian semigroup G(x) = (πz)− n 2 e −|x|2 z , (x ∈ R, 0), is O(|t|n2 ) , whereas the Poisson semigroup P (x) = Γ( 2 )π − 2 z(z + |x|2)−n+1 2 is O(|t| 2 ), on 1 + it as |t| → ∞, for n ≥ 2. From this we might expect that ‖P ‖1 would be O(1), as |t| → ∞, if n = 1 but, instead of this, we get O(log |t|), as is well known (see [S]). Moreover, there was no known example of a non-zero semigroup a in L1(R) bounded on 1 + it, (t ∈ R), and the same can be said for more general groups than R such as stratified Lie groups or non-compact groups with polynomial growth, for instance.