For \alpha a positive irrational, we consider the uniform subalgebra A_\alpha of C(T^2) consisting of those functions f satisfying \hat{f}(m,n)=0 whenever m+n\alpha<0. For positive irrationals \alpha, \beta, we determine when A_\alpha and A_\beta are isometrically isomorphic. Furthermore, we describe the group Aut(A_\alpha) of isometric automorphisms of A_\alpha. Finally we show how an explicit representation of Aut(A_\alpha) can be derived from Pell's equations.