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