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)

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. 

