Example 5

Let U be an open subset of $\mathbf{C}$ stable under complex conjugation. Consider the algebra $\mathscr{O}(U)$ of complex-valued holomorphic functions on U. For every function $f\in \mathscr{O}(U)$, the mapping $f^*:z\mapsto f(\overline{z})$ is a holomorphic function on U. The mapping $f\mapsto f^*$ is an involution on $\mathscr{O}(U)$.

Similarly, let S be a compact subset of $\mathbf{C}$ stable under complex conjugation. Consider the algebra $\mathscr{O}(S)$ of germs of complex-valued holomorphic functions in a neighbourhood of S. The mapping $f\mapsto f^*$ is an involution on $\mathscr{O}(S)$.

The subalgebra $\mathscr{O}_{\mathbf{R}}(U)$ (resp. the subalgebra $\mathscr{O}_{\mathbf{R}}(S)$) defined in n$^o13$ of I, p. 85 is the set of hermitian elements of $\mathscr{O}(U)$ (resp. of $\mathscr{O}(S)$).