Corollary 4.1.(i) in Johnstone's book Stone Spaces (electronic version: http://gen.lib.rus.ec/get?nametype=orig&md5=C26F62F69C32101307213F1960F85BA3)
states that the category of realcompact spaces is dual to the full subcategory
of the category of commutative rings consisting of rings of the form C(X).
The functor C implements the duality.

The category of compact CW-complexes embeds into the category of realcompact
spaces as a full subcategory, hence the functor C is fully faithful.