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Jul 18, 2011 at 19:06 comment added Matthew Daws Van Daele's algebraic framework is very nice, but it doesn't cover all locally compact groups-- you need Locally Compact Quantum Groups for that (see various papers by Kustermans and Vaes; also work of Woronowicz) and the associated Operator Algebraic machinery. Then you've replaced groups by algebras-- compact groups are just the unital algebras, and discrete groups the ones with compact dual. But this is maybe a long way from "unitary representations"...
Jul 18, 2011 at 19:02 history answered Andreas Thom CC BY-SA 3.0