Work on this project will concentrate on problems arising in the analysis of algebras generated by holomorphic or harmonic functions (function algebras). A metric is used to measure the distance between any two functions consisting of the maximum of the function value differences, often referred to as the uniform norm. The underlying theme of the work concerns algebras of functions which lie between familiar algebras. On a compact set in the plane one considers the effect of adjoining a function to a given function algebra (of holomorphic functions) and then completing the new collection into a larger algebra. A fundamental question is whether or not the new algebra consists of all continuous functions if the additional function was harmonic. This is resolved in a recent work if the harmonic function is real-valued. A clarification of the situation when it is complex-valued remains to be done. Some progress has also been made in the setting of several complex variables, where the same basic questions can be raised. At the present time, the domains are restricted to balls and the functions adjoined to the ball algebra of continuous holomorphic functions must have continuous first derivatives. A second goal of this work will be to relax the differentiability condition which appears to be unnecessary.
