Exercise 22, § 4

Let $X$ be a compact space, $A$ the normed algebra $C(X; \mathbf{C})$ over $\mathbf{C}$. For every closed ideal $a$ of $A$, prove that the relation $f \in a$ implies $\bar{f} \in a$ (remark that $\bar{f}$ is the uniform limit of functions of the form $gf$). Extend then to the algebra $A$ the results of Exercise 13 of $X$, p. 57.

¶ 23 a) Let $X$ be a compact space, $A$ a finite rank algebra over the field $\mathbf{R}$, admitting a unit element, and endowed with the topology defined in VI, p. 5. Let $B$ be a subalgebra of the $\mathbf{R}$-algebra $C(X; A)$; suppose that for every $\varepsilon > 0$, every pair of distinct points $x, y$ of $X$ and every pair of elements $u, v$ of $A$, there exists $f \in B$ such that $\| f(x) - u \| \leq \varepsilon$ and $\| f(y) - v \| \leq \varepsilon (\| z \|$ being any norm defining the topology of $A$); suppose moreover that if $u$ and $v$ belong to $\mathbf{R}$, there is a function $f \in B$ satisfying the preceding conditions and taking its values in $\mathbf{R}$. Under these conditions, prove that $B$ is dense in $C(X; A)$ for the topology of uniform convergence. (First prove that every function of $C(X; \mathbf{R})$ can be uniformly approximated by functions of $B$, then that the same is true of every constant $a \in A$, using the assumption and a partition of unity).

b) Take $A = \mathbf{H}$, field of quaternions over $\mathbf{R}$. Let $B$ be a subalgebra of the $\mathbf{R}$-algebra $C(X; \mathbf{H})$ such that: 1° for every function $f \in B$, the conjugate $\bar{f}$ of $f$ (defined by $\bar{f}(x) = f(\bar{x})$ for every $x \in X$) belongs to $B$; 2° for every $\varepsilon > 0$, every pair of distinct points $x, y$ of $X$ and every pair of elements $u, v$ of $\mathbf{H}$, there exists $f \in B$ such that $\| f(x) - u \| \leq \varepsilon$ and $\| f(y) - v \| \leq \varepsilon$. Prove that $B$ is everywhere dense in $C(X; \mathbf{H})$ (use $a$).

c) Extend to the algebra $C(X; \mathbf{H})$ the results of $X$, p. 57, Exercise 13, and now prove in particular that every closed ideal of this algebra is two-sided (reason as in Exercise 22 using $b$).

¶ 24) a) Let $X$ be a compact totally disconnected space, $A$ a topological ring. We equip the ring $C(X; A)$ with the topology of uniform convergence, which is compatible with the ring structure. Let $a$ be a left ideal of $C(X; A)$, and for every $x \in X$, let $F(x)$ be the closure in $A$ of the sets of the $f(x)$ for $f \in a$, which is a closed left ideal of $A$. Now prove that $\bar{a}$ is identical with the set of the $f \in C(X; A)$ such that $f(x) \in F(x)$ for every $x \in X$ (observe that for every $x \in X$ the neighbourhoods of $x$ which are both open and closed form a fundamental system of neighbourhoods of $x$).

b) Suppose in addition that there exists in $A$ a fundamental system of neighbourhoods of 0 which are two-sided ideals of $A$. Let $B$ be a subring of $C(X; A)$ containing the constants and separating the points of $X$; now prove that $B$ is dense in $C(X; A)$. (For every closed subset $F$ of $X$, every point $x \notin F$ and every neighbourhood $U$ of 0 in $A$, now prove that there exists $f \in B$ such that $f(x) = 1$ and $f(y) \in U$ for every $y \in F$).

25) Let $X$ be a locally compact space; we call strict topology on the vector space $C^b(X; \mathbf{R})$ of continuous and bounded numerical functions the topology defined by the family of distances $d_\varphi(f, g) = \sup_{x \in X} \varphi(x)|f(x) - g(x)|$, where $\varphi$ is a continuous numerical function, $\geq 0$ and tending to 0 at the point at infinity.

a) Now prove that for the uniform structure defined by these distances, $C^b(X; \mathbf{R})$ is complete and that the strict topology is compatible with the algebra structure on $\mathbf{R}$.

b) Now prove that if $A$ is a closed subalgebra of $C^b(X; \mathbf{R})$ for the strict topology, separating the points of $X$, and such that, for every $x \in X$, there exists $f \in A$ such that $f(x) \neq 0$, then $A = C^b(X; \mathbf{R})$.