Let G be an nonAbelian locally compact group. What is the set af all multiplicative functionals of L1(G)? (When G is abelian the answer is the dual group)
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Multiplicative functionals on the group algebra correspond to onedimensional unitary representations of the group. In the noncommutative case such (nontrivial) representations can be absent or constitute a small part of the set of all irreducible unitary representations. See, for example, M. A. Naimark, Normed rings, Noordhof, Groningen, 1970. 

