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)
1 Answer
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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.
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$\begingroup$ Just to expand on the word "nontrivial" - there is of course at least one multiplicative functional on $L^1(G)$, namely the augmentation character $f\mapsto \int_G f$. The point is that there may not be any others. $\endgroup$ Apr 21, 2011 at 8:46
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