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Daniel Turcotte - Propagation of involutive properties of analytic functions with values in complex unital Banach algebras with involutions
DANIEL TURCOTTE, Ryerson Polytechnic University, Toronto, Ontario M5B 2K3, Canada | |
Propagation of involutive properties of analytic functions with values in complex unital Banach algebras with involutions |
Let , where denotes the set of -valued analytic functions from an open set of the complex plane and is a complex unital Banach algebra with involution. We note that in general the function is not analytic. However, we show that the principle of analytic continuation can still be applied in a restricted way to to obtain a new principle of -analytic continuation along regular analytic Jordan arcs.
Theorem. Let . Let be a regular analytic Jordan arc contained in and (sn) a converging sequence contained in such that for . If there exists an analytic -valued function l from an open neighborhood of such that for every sn, then for every .
Corollary. Let . Let be a regular analytic Jordan arc contained in and (sn) a converging sequence contained in such that for . If f(sn) is unitary (resp. normal, self-adjoint, anti-self-adjoint) for every sn then f(s) is unitary (resp. normal, self-adjoint, anti-self-adjoint) for every .
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