  A pro-C^*-algebra is a (projective) limit of C^*-algebras in the category of
topological *-algebras. From the perspective of non-commutative geometry,
pro-C^*-algebras can be seen as non-commutative k-spaces. An element of a
pro-C^*-algebra is bounded if there is a uniform bound for the norm of its
images under any continuous *-homomorphism into a C^*-algebra. The *-subalgebra
consisting of the bounded elements turns out to be a C^*-algebra. In this
paper, we investigate pro-C^*-algebras from a categorical point of view. We
study the functor (-)_b that assigns to a pro-C^*-algebra the C^*-algebra of
its bounded elements, which is the dual of the Stone-\v{C}ech-compactification.
We show that (-)_b is a coreflector, and it preserves exact sequences. A
generalization of the Gelfand-duality for commutative unital pro-C^*-algebras
is also presented.
