Let $C(\mathbb{R};{U}(n))$ denote the topological group of continuous functions $\mathbb{R}\to {U}(n)$ with pointwise multiplication and compact-open topology. My question is:

**Are these groups isomorphic for different values of $n$?**

I suspect the answer is no (it feels like it should be obvious), but proving this in the category of topological groups seems difficult. I would *like* to associate to $C(\mathbb{R}; {U}(n))$ the enveloping C*-algebra $C_b(\mathbb{R};M_n)$, since these are much easier to distinguish$^1$. So a second question would be:

**Does there exist a functor $TopGrp\to C^*Alg$ taking $C(\mathbb{R};{U}(n))$ to $C_b(\mathbb{R};M_n)$?**

The same question could be asked of the measure theoretic versions of these groups $\mathcal{M}(\mathbb{R};{U}(n))$. These are called current groups, although the literature seems unhelpful for the isomorphism problem.

$_{^1\text{ e.g. looking at Murray-von Neumann equivalence classes of projections will distinguish them.}}$