Example 5

Let Δ be the unit disc of complex numbers $z$ such that $|z|\leqslant 1$. Its boundary in $\mathbf{C}$ is $\mathbf{U}$. Let A be the Banach algebra of complex functions $f$ on $\mathbf{U}$ such that there exists a continuous function $\widetilde{f}\in \mathscr{C}(\Delta )$ extending $f$ which is analytic in $\mathring{\Delta}$, endowed with the norm $\|f\|=$ sup$_{z\in\mathbf{U}}|f(z)|$. By virtue of the maximum principle (VAR, R1, p. 30, 3.3.7), we then have $\|f\|=$ sup$_{z\in\Delta}|\widetilde{f}(z)|$, and therefore A coincides with the algebra of example 9 of I, p. 20. The set $\mathsf{X}(A)$ is identified with Δ and, if $f\in A$, the mapping $\mathscr{G}(f)$ is identified with the continuous extension of $f$ to Δ which is analytic in $\mathring{\Delta}($cf. exerc. 6 of I, p. 193).