Does smoothness of left and right multiplication imply smoothness of multiplication?

differential-geometry differential-topology lie-groups topological-groups smooth-manifolds

Solution 1:

It is proven in

Ellis, Robert, Locally compact transformation groups, Duke Math. J. 24, 119-125 (1957). ZBL0079.16602.

that a locally compact Hausdorff topological space equipped with a group structure where both left and right multiplications are continuous, is actually a topological group, i.e. the multiplication is jointly continuous. Now, you can use the result that a topological manifold which is also a topological group admits a unique Lie group structure.

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