Suppose $X$ is a compact metric space. Is there a good candidate for a dense subalgebra $A\subseteq C(X)$, such that the inclusion induces an isomorphism in $K$-theory? For example, if $X$ was a manifold, I could take $A$ to be the smooth functions. Would it help if $X$ was a finite CW-complex?

P.S.: I know that $A=C(X)$ is a good candidate :)

EDIT: In response to a comment below, let me add a few comments. The problem with $C(X)$ is that, as any other nuclear $C^*$-algebra, it has trivial higher (continuous) cyclic cohomology, therefore we don't get interesting pairings with the corresponding homology. Ideally, I would like $A$ to be such that

- It has interesting cyclic cohomology
- The (even, odd) periodic cyclic homology of A coincides with (even, odd) Cech, or Alexander-Spanier, cohomology of $X$.