Let $G$ and $H$ be two compact Lie groups with isomorphic Lie algebras $\frak{h} \simeq \frak{g}$, but which are non-isomorphic as topological spaces. From the isomorphism assumption it (should) follows that we have a bijection between the irreducible representations of $G$ and $H$. From this it should also follow that the Peter–Weyl decomposition of the space matrix coefficient functions of both groups are isomorphic as $\frak{g}$-modules. Now the completion of the space of matrix coefficients of both groups to their spaces of continuous functions cannot be isomorphic since $G$ and $H$ are not isomorphic as topological spaces. So where does this failure of isomorphism come from? Does it come from the algebra structure of the space of matrix coefficients? Or might it come from the completion being different, i.e. the $\|\cdot\|_{\infty}$ norm is different in both cases.

Edit: Following the comments below, the example I am thinking about is $U_2$ and $SU_2 \times U_1$. Does $SU_2 \times U_1$ have more irreducible representations than $U_2$?

groups"? "Isomorphism as topologicalspaces" is a strange property to study for Lie groups. (But, come to that, I don't actually know whether this is one of those instances of rigidity of Lie groups where homeomorphic Lie groups are automatically isomorphic Lie groups.) $\endgroup$ – LSpice Dec 22 '20 at 19:43