The unit group of a non-discrete irreducible continuous ring admits no non-trivial unitary representations continuous with respect to the strong operator topology.
Kirillov,Positive-definite functions on a group of matrices with elements from a discrete field
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Unitary representations and von Neumann's continuous geometries
The unit group of a non-discrete irreducible continuous ring admits no non-trivial unitary representations continuous with respect to the strong operator topology.