Exercise 1, § 8

Let $X$ be a topological space, $\mathscr{C}_b(X)$ the star-algebra of bounded continuous complex functions on $X$ and $\widehat{X} =\mathsf{X}(\mathscr{C}_b(X))$.

a) Let $A$ be a unital Banach subalgebra of $\mathscr{C}_b(X)$ and let $R_A$ be the equivalence relation in X such that $x$ is equivalent to $x'$ if $f(x) =f(x')$ for every function $f\in A$. The saturated subset S for $R_A$ of a compact subset K of X is closed. (Let $x\in \overline{S}$; for every $f\in A$ and every $\varepsilon >0$, let $V_{f,\varepsilon}$ be the set of the $x'\in X$ such that $|f(x)-f(x')|\leqslant \varepsilon$; if $f_1, . . . , f_n\in A$ and $\varepsilon_1, . . . , \varepsilon_n>0$, one has

therefore

$\varepsilon >0$

whence $x\in S$.)

b) Suppose X is normal. Let R be a closed equivalence relation in X and let $A = A_R$ be the subalgebra of $\mathscr{C}_b(X)$ formed by the functions constant on the classes according to R. Now prove that $R = R_A$. (Use TG, IX, p. 105, exerc. 19, a).)

c) Suppose X is compact. Then $R\mapsto A_R$ is a bijection from the set of separated equivalence relations of X onto the set of star-algebras of $\mathscr{C}(X)$. (Let A be a star-algebra of $\mathscr{C}(X)$. Now prove that $R_A$ is separated, and apply the Weierstrass-Stone theorem to the algebra of continuous functions on $X/R_A$ deduced from A by passing to the quotient.)