Let $M$ be a compact spin$^c$ manifold, so that it has a fundamental class $[M] \in K_n(M)$. It is well-known that the cap product with $[M]$ induces Poincare duality isomorphisms $K^\ast(M) \cong K_\ast(M)$ between $K$-theory and $K$-homology.

What about the non-compact case? Do we have $K^\ast_{\text{cpt}}(M) \cong K_\ast(M)$ via the cap product with $[M]$, where $K_\ast(M)$ now stands for the $K$-homology of $C_0(X)$? Any references / proofs?

I'm wondering, since this is never mentioned anywhere.