Let G be an non-Abelian 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 one-dimensional unitary representations of the group. In the non-commutative 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.