  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 of dimension at least three; and an essential, natural uniform algebra
on the unit sphere in C^3 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.
