Corollary 3

Let A be a commutative unital star algebra.

a) There exists a compact topological space X such that A is isomorphic to the star algebra $\mathscr{C}(X)$;

b) Let B be a commutative unital star algebra. The mapping $\pi \mapsto \mathsf{X}(\pi )$ is a bijection of the set of unital morphisms of star algebras $A\rightarrow B$ onto the set of continuous mappings of $\mathsf{X}(B)$ into $\mathsf{X}(A)$.

This follows from the foregoing and from prop. 4 of I, p. 35.