Abstract

A uniform algebra A on its Shilov boundary X is maximal if A is not C(X) and there is no uniform algebra properly contained between A and C(X). It is essentially pervasive if A is dense in C(F) whenever F is a proper closed subset of the essential set of A. If A is maximal, then it is essentially pervasive and proper. We explore the gap between these two concepts. We show the following: (1) If A is pervasive and proper, and has a nonconstant unimodular element, then A contains an infinite descending chain of pervasive subalgebras on X. (2) It is possible to imbed a copy of the lattice of all subsets of N into the family of pervasive subalgebras of some C(X). (3) In the other direction, if A is strongly logmodular, proper and pervasive, then it is maximal. (4) This fails if the word ‘strongly’ is removed. We discuss further examples, involving Dirichlet algebras, A(U) algebras, Douglas algebras, and subalgebras of H1(D). We develop some new results that relate pervasiveness, maximality and relative maximality to support sets of representing measures.