Banach Algebras where the Singular Elements Are Removable Singularities

Let A be a CU-algebra with identity and suppose A has real rank 0. Suppose a complex-valued function is holomorphic and bounded on the intersection of the open unit ball of A and the identity component of the set of invertible elements of A. We show that then the function has a holomorphic extension to the entire open unit ball of A. Further, we show that this does not hold when AsC S., where S is any compact Hausdorff space that contains a homeomorphic image of the intervalw0, 1x.