Banach algebras where the singular elements are removable singularities

Let A be a C*-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 invertibl e 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 hol d when A = C(S), where S is any compact Hausdorff space that contains a hom eomorphic image of the interval [0, 1]. (C) 2000 Academic Press.