NB the original question asked about $\beta\mathbb{D}$ rather than the corona, hence some of the initial comments.




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Is there a group operation on $\beta \mathbb{D} \setminus \mathbb{D}$ extending complex multiplication on $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$? 



Is $\beta \mathbb{D}\setminus \mathbb{D}$ a group?
No, because $\beta\mathbb D$ and $\beta\mathbb D\setminus \mathbb D$ are not Dugundji compact (and not supercompact) whereas all compact topological groups are Dugundji compact according to a result of Uspenskii
<https://www.researchgate.net/publication/266010994_Topological_groups_and_Dugundji_compacta>
and supercompact (by a result of Mills reproved by Kubis and Turek in <http://link.springer.com/article/10.2478/s11533-011-0019-x>).


Another reason why $\beta \mathbb D$ or $\beta\mathbb D\setminus\mathbb D$ cannot possess a group structure with continuous left (or right) shifts is that those spaces are not topologically homogeneous (because of the existence of weak P-points, for example).


