Exercise 19, § 1

Let X be a compact space, A a normed algebra over $\mathbf{R}$ with unity element, and G a compact group; assume that G operates continuously on A and on X and that, for every $s \in G$, $a \mapsto s \cdot a$ is an automorphism of the algebra A such that $\|s \cdot a\| = \|a\|$ for all $a \in A$. A mapping $f$ of X into A is said to be covariant under G if

for all $s \in G$ and $x \in X$. Let B be a subring of $A^X$ consisting of covariant continuous functions and containing all of the covariant continuous mappings of X into $\mathbf{R}$ ($\mathbf{R}$ being identified with a subalgebra of A; one observes that for such a mapping $g$, $g(s \cdot x) = g(x)$ for all $s \in G$ and $x \in X$). Let $f$ be a covariant continuous mapping of X into A, and assume that for every $y \in X$, there exists a mapping $g_y \in B$ such that $f(y) = g_y(y)$. Then, for every $\varepsilon > 0$, there exists a mapping $g \in B$ such that $\|f(x) - g(x)\| \leq \varepsilon$ for all $x \in X$. (Make use of a suitable continuous partition of unity $(\varphi_i)$ on X and introduce the functions $h_i$ such that $h_i(x) = \int_G \varphi_i(s \cdot x)\, ds$.)