Relation between $\text{Hom}_{\mathsf{Alg}_{\mathbb{R}}}(\mathcal{C}^\infty(M),A) $ and $ X \otimes_\mathbb{R} A$?

Question

This question is a little bit of a shot in the dark, but maybe someone stumbled over it before... Let $M$ be a (simply connected) smooth manifold modelled on a locally convex space $X$ over $\mathbb{R}$. Moreover, let $A$ be a fin. dim. commutative $\mathbb{R}$-algebra. Is there an (at least on sets) isomorphism between $\text{Hom}_{\mathsf{Alg}_{\mathbb{R}}}(\mathcal{C}^\infty(M),A) $ and $ X \otimes_\mathbb{R} A$? Maybe when substituting X by its dual? I think this is possible for fin. dim. manifolds, but i have no idea what happens in the general case...

Answer 1

No. If $A=\mathbb{R}$ and $X=\mathbb{R}^n$ (i.e. $M$ is a usual smooth $n$-manifold), then it is known that $\hom(C^{\infty}(M),\mathbb{R}) = \{\mathrm{ev}_p : p \in M\}$ (where $\mathrm{ev}_p(f):=f(p)$), and this has no natural bijection to $\mathbb{R}^n$. (In fact, no bijection when $M$ is empty.)

A more meaningful question would be: Is $\hom(C^{\infty}(M),\mathbb{R}) \otimes_{\mathbb{R}} A \to \hom(C^{\infty}(M),A)$, $\alpha \otimes a \mapsto (p \mapsto \alpha(p) a)$ an isomorphism? (I don't know the answer.)