Corollary 2

Let A be a commutative star algebra.

a) There exists a locally compact topological space X such that A is isomorphic to the star algebra $\mathscr{C}_0(X)$;

b) Let B be a commutative star algebra. The mapping $\pi \mapsto$ $\mathsf{X}'(\pi )$ is a bijection of the set of morphisms of star algebras of A into B onto the set of proper partial mappings of $\mathsf{X}(B)$ into $\mathsf{X}(A)$ (def. 1 of I, p. 33).

Theorem 1 establishes the first assertion, and the second assertion follows from prop. 3 of I, p. 34 and from example 2 of I, p. 102.