Remark

*Let $\mathbf{G}$ be the category whose objects are the locally compact spaces and whose morphisms are the proper partial mappings (def. 1 of I, p. 33), and let $\mathbf{S}$ be the category of commutative star algebras, whose morphisms are the morphisms of involutive algebras. Consider the functor from $\mathbf{S}$ into the opposite category $\mathbf{G}^{\circ}$ which associates with a commutative star algebra A the locally compact space $\mathsf{X}(A)$ of nonzero characters of A, and with a morphism $\varphi : A\rightarrow B$ of commutative star algebras the continuous mapping $\mathsf{X}'(\varphi )$. Th. 1 and cor. 2 mean that this functor is an equivalence of categories, and that a quasi-inverse of this functor is the functor which associates with a locally compact topological space X the commutative star algebra $\mathscr{C}_0(X)$.

Analogously, corollary 3 means that the opposite category of the category of compact spaces is equivalent to the category of commutative unital star algebras.*