Function algebras on Riemann surfaces and Banach spaces

by Boos, Lynette J.

ii Alexander J. Izzo, advisor For K a compact subset of a Riemann surface, necessary and sufficient conditions are given for a function algebra containing A(K) to be all of C(K). Using these results, several conditions are given on a complex-valued function f so that the algebra generated by A(K) and f is all of C(K). In particular, the results are applied to a harmonic function f to give sufficient conditions for the algebra generated by A(K) and f to be all of C(K). Also, sufficient conditions are given for the algebra A(K) to be a maximal subalgebra of C(K). For X a compact subset of a Banach space, six properties that a compact subset K of the boundary of X can have in relation to the algebra A(X) are considered. These properties include the concepts of totally null set, zero set and null set. In the case of the finite and infinite dimensional polydiscs it is shown that five of the properties are equivalent, and a counterexample is given to show that the property of being a totally null set is weaker than the other five properties. This is in contrast to the unit ball of Cn, where all six properties are known to be equivalent. The construction of the counterexample is then modified to give a condition on an algebra such that the property of being a totally null set is weaker than the property of being a null set. Finally, several conditions on an algebra are given, each of which implies that the property of being a totally null set is equivalent to the property of being a null set.