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1. Noncompact group G doesn't have faithfull (ie. kernel has more that one element) unitary representation.

Proof:

If D(G) is faithfull unitary representation of group G, then D(G) is closed (in topological sense) subgroup of group U(n) (unitary group which is compact and connected, so every it's subgroup is closed), so D(G) is compact (closed subset of compact set is compact), and because D(G) is faithfull we conclude that G is also compact.

2. Kernel of smooth homomorphism is discrete subgroup or whole group.

Proof:

If we denote group with G and kernel with K, K is subgroup. If indentiry component has more than one element than we see that open set containing indentity is represented with identity matrix. Because whole group is genereated with elements from around identiry (my english is bad, but I hope you understand) so whole group G is represented with unitary matrix so K=G. I other case K is discerete subgroup.

And I have few more statements that I doesn't know how to prove nor if they are correct:

3. Semisimple, noncompact group G doesn't have unitary representation.

4. Orthogonal complement (in sense of Killing form) of kernel of representation of algebra doesn't have intersection with kernel (except zero).

If someone sees mistakes here or know for sure that some statements are incorrect, please let me know.