Isomorphism of uniform algebras on the 2-torus

TL;DR

This paper investigates the isomorphism classes of certain uniform algebras on the 2-torus parameterized by irrational numbers, determining when they are equivalent and describing their automorphism groups.

Contribution

It provides a classification of the algebras lphay their isomorphism types and explicitly characterizes their automorphism groups, filling a gap in the understanding of these algebras.

Findings

lphare not isomorphic for different irrationals lphand etaxcept in trivial cases.

The automorphism group of lphare explicitly determined.

The classification distinguishes lpharom earlier properties like maximality and Gelfand space.

Abstract

For $\alpha$ a positive irrational, let $\mathcal{A}_{\alpha}$ be the subalgebra of continuous functions on the two-torus whose Fourier transform vanishes at $(m, n)$ if $m + \alpha n < 0.$ These algebras were studied by Wermer and others, who proved properties such as maximality and characterized the Gelfand space. One of the major themes of current work in operator algebras is classification, but none of the properties which were investigated earlier distinguished between $\mathcal{A}_{\alpha}$ and $\mathcal{A}_{\beta},$ if $\beta $ is another positive irrational. We address this question. We also determine the automorphism group of $\mathcal{A}_{\alpha}.$