Type I group

Summary

A locally compact group is of type I when every factor representation generates a type I factor. Then its unitary representations break up into irreducibles in an essentially unique way. For such groups the unitary dual is a well-behaved space, and the Plancherel theorem decomposes the regular representation as a direct integral of irreducibles. Abelian groups, compact groups, connected semisimple Lie groups and connected nilpotent Lie groups are type I groups. Non-abelian free groups are not, and for them the decomposition into irreducibles is far from unique.