TITLE: Naive question on adelic groups
QUESTION [6 upvotes]: The ever-reliable Wikipedia says:

... an adelic algebraic group is a semitopological group defined by...

No more details are given, and I was wondering if the multiplication only being separately continuous has any noticeable effect when working with adelic groups. I ask because there are some algebras which pop up in the study of von Neumann algebras where the multiplication is not jointly continuous, and this is mildly annoying.

REPLY [8 votes]: A locally compact semitopological semigroup which is also a group is in fact a topological group, i.e. the existence of inverses plus separate continuity actually forces joint continuity. 
This is a theorem of Ellis, see

R. Ellis, Locally compact transformation groups.
Duke Math. J. 24 (1957) no. 2, 119–125

Quoting from the MathReview:

Let G be a group of homeomorphisms on a locally compact space X. Suppose G has a Hausdorff topology such that multiplication is continuous in each variable separately. If the function π:G × X → X defined by π(g,x)=g(x) is continuous on the left, the author shows that π is jointly continuous.

EDIT: it's been noted in the comments that the part I quoted does not explain in any way why we can deduce continuity of inversion. In an earlier paper Ellis had shown that in a topological semigroup whose underlying semigroup is a group, inversion is continuous:

R. Ellis, A note on the continuity of the inverse. Proc. Amer. Math. Soc. 8 (1957), 372–373