A Weil Banach Algebra for Multiplicative Algebraic Numbers

Abstract

We present a generalization of the construction of Allcock and Vaaler in [AV09] which established that the Weil height provides non-zero algebraic numbers modulo torsion with the structure of a normed vector space and construct its completion as L1 of a locally compact measure space. In the present work we show that this space is naturally a Banach algebra and describe its structure and a method for computing the Banach multiplication explicitly using the theory of harmonic analysis on compact groups.