Corollary 1

Let A be a unital star algebra and let $A_n$ be the set of normal elements of A. Endow the space $\mathscr{C}(\mathbf{C})$ with the topology of compact convergence. The mapping $(f, a)\mapsto f(a)$ of $\mathscr{C}(\mathbf{C})\times A_n$ into A is continuous.

Let $(f_0, a_0)\in \mathscr{C}(\mathbf{C})\times A_n$. Let U be a relatively compact neighbourhood of the spectrum of $a_0$. Let V be the set of normal elements $a$ of A such that Sp$_A(a)\subset U$; this is an open neighbourhood of $a_0$ in $A_n$ (I, p. 76, Prop. 10). For every $a\in V$ and every function $f\in \mathscr{C}(\mathbf{C})$, one has $f(a) = (f|U)(a)$. Since the mapping $f\mapsto \|f|U\|_{\infty}$ is a continuous seminorm on $\mathscr{C}(\mathbf{C})$ endowed with the topology of compact convergence, the continuity of the mapping $(f, a)\mapsto f(a)$ at $(f_0, a_0)$ follows from Prop. 10.