Concerning Function Algebras.
This paper is devoted to an investigation of collections of functions that satisfy a condition similar to the hypothesis of Morera's theorem. We start with a function g that is continuous on the closed unit disk, and of hounded variation on each simple closed rectifiable curve C contained in the open unit disk U. We denote by A(g), the collection of all complex-valued simple closed rectifiable curve C contained in U. A(g) , with the usual pointwise addition and scalar multiplication of functions, is a Banach space under supremum norm. One of our main concerns is the study of conditions under which A(g) is closed under pointwise multiplication,and hence becomes a Banach algebra. In Chapter I, we impose conditions such that A(g) becomes an algebra. In three cases, A(g) is then identified with an algebra of analytic functions. These conditions are, respectively, that A(g) contain a nonconstant analytic function, that g be one-to-one and that g ̂ be of bounded variation on simple closed rectifiable curves, and that A(g) contain such a function. In the fourth case considered, A(g) can be identified with all complex-valued continuous functions on an arc. In Chapter II, we develop some of the general theory continuous functions on (T such that C 0 for each