Removable Singularities In
نویسندگان
چکیده
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 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 A = C(S), where S is any compact Hausdorr space that contains a homeomorphic image of the interval 0,1]. Throughout, A denotes a Banach algebra with identity e, A inv denotes the open set of invertible elements of A, and A e inv denotes the identity component of A inv. Also, A 0 denotes the open unit ball of A, i.e., A 0 = fx 2 A : kxk < 1g; and D(A) denotes A 0 \ A e inv. In particular, D(C) is the open unit disc punctured at the origin. Definition 1. A Banach algebra A has the removable singularity property if every bounded holomorphic function f : D(A) ! C has a holomor-phic extension to A 0. In this deenition we consider the identity component of A inv rather than A inv since if A inv is not connected the holomorphic function on A inv which is 1 on one of the components and 0 on the others has no holomorphic extension to A 0. It follows from Theorem 3 given in the next section (or see 12, Theorem 3]) that our deenition is no more stringent when the range of f is allowed to be any Banach space.
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