Exercise 6, § 8

Let Δ be the disk $|\zeta |\leqslant 1$ in $\mathbf{C}$ and A the Banach algebra of continuous complex-valued functions on Δ analytic in the interior of Δ (example 9 of I, p. 20).

a) If $x\in A$, then $x$ is the limit in A of the functions

The identity mapping of Δ is a generator of A.

b) The space $\mathsf{X}(A)$ is canonically identified with Δ. (Use assertion g) of the prop. 1 of I, p. 142.)

c) Let S be a closed subset of Δ, identified with a closed subset of $\mathsf{X}(A)$. If S possesses a non-isolated point $\zeta$ such that $|\zeta |<1$, the closure of S for the Jacobson topology is Δ.

d) For $x\in A$, put $x^*(\zeta ) =\overline{x(\overline{\zeta})}$. Then $x\mapsto x^*$ is an isometric involution of A. The hermitian characters of A are defined by the real elements of Δ .

e) The mapping $x\mapsto x|\mathbf{U}$ is an isomorphism of A onto $P(\mathbf{U})$. Every function of $\mathscr{C}_{\mathbf{R}}(\mathbf{U})$ is a uniform limit of real parts of functions of $P(\mathbf{U})$.