Exercise 20, § 4

a) Let $X$ be a compact space, let $A$ be an $\mathbf{R}$-algebra of finite rank with an identity element, and endow $A$ with the topology defined in Chapter VI, § 1, no. 5. Let $B$ be a subalgebra of the $\mathbf{R}$-algebra $\mathscr{C}(X; A)$, and suppose that, for each $\varepsilon > 0$, each pair of distinct points $x, y$ of $X$ and each pair of elements $u, v$ of $A$, there exists $f \in B$ such that $\|f(x) - u\| \leq \varepsilon$ and $\|f(y) - v\| \leq \varepsilon$ ($\| \ \|$ being any norm defining the topology of $A$); and suppose, further, that if $u$ and $v$ belong to $\mathbf{R}$, there is a function $f \in B$ satisfying these conditions and taking its values in $\mathbf{R}$. Show that, under these hypotheses, $B$ is dense in $\mathscr{C}(X; A)$ with respect to the topology of uniform convergence. [Show first that every function of $\mathscr{C}(X; \mathbf{R})$ can be uniformly approximated by functions of $B$ and then that the same is true of every constant $a \in A$, by using the hypotheses and a partition of unity.]

b) Take $A = \mathbf{H}$, the division ring of quaternions over $\mathbf{R}$. Let $B$ be a subalgebra of the $\mathbf{R}$-algebra $\mathscr{C}(X; \mathbf{H})$ such that: (i) for each $f \in B$, the conjugate $\overline{f}$ of $f$ [defined by $\overline{f}(x) = \overline{f(x)}$ for all $x \in X$] belongs to $B$; (ii) for each $\varepsilon > 0$, each pair of distinct points $x, y$ of $X$ and each pair of elements $u, v$ of $\mathbf{H}$, there exists $f \in B$ such that $\|f(x) - u\| \leq \varepsilon$ and $\|f(y) - v\| \leq \varepsilon$. Show that $B$ is dense in $\mathscr{C}(X; \mathbf{H})$ [use a)].

c) Extend the results of Exercise 13 to the algebra $\mathscr{C}(X; \mathbf{H})$, and show in particular that every closed ideal of this algebra is two-sided [argue as in Exercise 19, using b)].