Notes on a special order on ${\mathbb{Z}}^{\infty }$

Abstract

In 1958, Helson and Lowdenslager extended the theory of analytic functions to a general class of groups with ordered duals. In this context, analytic functions on such a group $G$ are defined as the integrable functions whose Fourier coefficients lie in the positive semigroup of the dual of $G$. In this paper, we found some applications of their theory to infinite-dimensional complex analysis. Specifically, we considered a special order on ${\mathbb{Z}}^{\infty }$ and corresponding analytic continuous functions on ${\mathbb{T}}^{\ddot{I}}‰$, which serves as the counterpart of the disk algebra in infinitely many variables setting. By characterizing its maximal ideals, we have generalized the following theorem to the infinite-dimensional case: For a positive function $w$ that is integrable and log-integrable on ${\mathbb{T}}^d$, there exists an outer function $g$ such that $w=|g{|}^2$ if and only if the support of $\widehat{\log w}$ is a subset of ${\mathbb{N}}^d\cap (-\mathbb{N}{)}^d$. Furthermore, we have found the counterpart of the above function algebra in the closed right half-plane, and the representing measures of each point in the right half-plane for this algebra. As an application of the order, we provided a new proof of the infinite-dimensional Szegö's theorem.

This version revises the proof main theorem (Theorem 6.1) due to a fundamental error identified in the original manuscript