The contravariant functor $C()$ given by $$ \hom_{Top}(,\mathbb{R}):cCW\to Rng $$ where $cCW$ is the category of compact CW complexes is injective on objects. What is known about surjectivity, faithfulness and fullness of this functor?

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 CWcomplexes embeds into the category of realcompact spaces as a full subcategory, hence the functor C is fully faithful. 

