Suppose $K$ is a compact topological Hausdorff space with a dense subspace $G$.
Moreover, let $G$ have a group structure which is compatible with the topology inherited from $K$. i.e. $G$ is a topological group with the subspace topology.
Is it known under what circumstances will ensure that the multiplication can be extended continuously to all of $K$? or does the answer depend on the group?
For a simple counter example, take $G=(0,1)$ and $K=[0,1]$. Then, $G$ is homeomorphic to $(0,\infty)$ which has a multiplicative group structure that pulls back to $G$ as $$m:G\times G\to G,\quad m(x,y)=\left(\frac{1}{x}-1\right)\left(\frac{1}{y}-1\right).$$ Now it is clear that $m$ does not extend to $K$.