Daniel Turcotte - Propagation of involutive properties of analytic functions with values in complex unital Banach algebras with involutions

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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 $f \in {\rm H}(\Omega,{\bf B})$ , where ${\rm H}(\Omega,{\bf B})$ denotes the set of ${\bf B}$ -valued analytic functions from an open set $\Omega$ of the complex plane and ${\bf B}$ is a complex unital Banach algebra with involution. We note that in general the function $s\mapsto f(s)^\ast$ is not analytic. However, we show that the principle of analytic continuation can still be applied in a restricted way to $f(s)^\ast$ to obtain a new principle of $\ast$ -analytic continuation along regular analytic Jordan arcs.

Theorem. Let $f \in {\rm H}(\Omega,{\bf B})$ . Let $\Gamma$ be a regular analytic Jordan arc contained in $\Omega$ and (s_n) a converging sequence contained in $\Gamma$ such that $s_n\not=s_m$ for $n\not=m$ . If there exists an analytic ${\bf B}$ -valued function l from an open neighborhood of $\Gamma$ such that $f(s_n)^\ast=l(s_n)$ for every s_n, then $f(s)^\ast=l(s)$ for every $s\in \Gamma$ .

Corollary. Let $f \in {\rm H}(\Omega,{\bf B})$ . Let $\Gamma$ be a regular analytic Jordan arc contained in $\Omega$ and (s_n) a converging sequence contained in $\Gamma$ such that $s_n\not=s_m$ for $n\not=m$ . If f(s_n) is unitary (resp. normal, self-adjoint, anti-self-adjoint) for every s_n then f(s) is unitary (resp. normal, self-adjoint, anti-self-adjoint) for every $s\in \Gamma$ .

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