Remark

*In the language of category theory, the preceding results may be interpreted in the following way. Let $\mathbf{G}$ be the category whose objects are locally compact topological spaces and whose morphisms are proper partial mappings. The functor X $\mapsto \mathscr{C}_0(X)$ is a contravariant, fully faithful functor from the category $\mathbf{G}$ into the category of commutative complex Banach algebras. Moreover, $A\mapsto \mathsf{X}(A)$ is a contravariant functor from the category of commutative complex Banach algebras into the category $\mathbf{G}$. If to a locally compact topological space X one associates the homeomorphism

ev$: X\rightarrow \mathsf{X}(\mathscr{C}_0(X))$,

one obtains an isomorphism of the identity functor of the category $\mathbf{G}$ onto the composite functor $X\mapsto \mathsf{X}(\mathscr{C}_0(X))$.

It is not true that the composite functor $A\mapsto \mathscr{C}_0(\mathsf{X}(A))$ is isomorphic to the identity functor of the category of commutative complex Banach algebras (cf. example 2 of I, p. 36 and exercise 2 of I, p. 155). One will, however, see a statement of this type for commutative star-algebras (No. 5 of I, p. 107).*