Exercise 16, § 6

Let X be a topological space. Let $\mathscr{C}_b(X)$ be the commutative stellar algebra of bounded continuous complex-valued functions on X.

a) Let $\widehat{X} =\mathsf{X}(\mathscr{C}_b(X))$ ; the space $\widehat{X}$ is a compact space such that $\mathscr{C}(\widehat{X})$ is identified with $\mathscr{C}_b(X)$.

b) The mapping $j: X\rightarrow \widehat{X}$ which sends $x$ to the character $f\mapsto f(x)$ is continuous.

c) If X is completely regular (TG, IX, p. 8, def. 4), the mapping $j$ is a homeomorphism of X onto a dense subspace of $\widehat{X}$. (To show that the image of $j$ is dense in $\widehat{X}$, show that a continuous complex function on $\widehat{X}$ which is zero on $j(X)$ is identically zero.)

d) The space $\widehat{X}$ is identified with the Stone–Čech compactification of X (TG, IX, p. 9).