A general method for constructing essential uniform algebras with prescribed properties is presented. Using the method, the following examples are constructed: an essential, natural, regular uniform algebra on the closed unit disc; an essential, natural counterexample to the peak point conjecture on each manifold-with-boundary of dimension at least three; and an essential, natural uniform algebra on the unit sphere in C3 containing the ball algebra and invariant under the action of the 3-torus. These examples show that a smoothness hypothesis in some results of Anderson and Izzo cannot be omitted.
Feinstein, J. F., & Izzo, A. J. (2019). A general method for constructing essential uniform algebras. Studia Mathematica, 246, 47-61. https://doi.org/10.4064/sm170907-23-2