algebra; the group r becomes the boundary of the disc, and all functions take their maximum modulus on r. For interior points r of the disc, there is a measure on r such that the value at r is given by an integral of the boundary values. This "Poisson integral" is studied in detail, and it is shown that one of these "generalized holomorphic" functionsf cannot vanish on an open set of the boundary which has positive measure with respect to all of these harmonic measures. Moreover, when G has sufficiently many continuous "characters" (with values in the unit disc), and f vanishes on a nonvoid open set, then f =0. In particular, the algebra becomes an integral domain. We can thus make arbitrarily huge (semi-simple) Banach algebras which are integral domains. We wish to cite some work related to ours done by Goldman [VI] and by Mackey [VII]. Goldman states 3.1 when G is a totally ordered discrete group. Moreover, our 4.6 was motivated by a discussion with him concerning possible generalizations of the classical case of the disc. Mackey also states 3.1 and has a complicated Cauchy integral formula. His point of view is much