Isomorphism of uniform algebras on the 2-torus

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 $\mathcal{A}_{\alpha}.$