Given a commutative operator algebra, when is its C*-envelope commutative as well? Motivated by classification problems in operator theory, we study this question for quotients of certain operator algebras of holomorphic functions on the complex unit ball. In one variable, the answer is negative in all non-trivial situations. In the multivariate case however, the picture is more complicated. Nevertheless, we show how the presence of isolated points in the character space of the quotient often guarantees that the C*-envelope is not commutative. This is joint work with Edward Timko.