Cartan subalgebras, transverse measures and non-type-I Plancherel formulae

Two formulations are given of Plancherel Theory for locally compact second countable groups. The first applies to arbitrary such groups, and recovers the L²-norm of suitable functions on the group from data associated to the central decomposition of the regular representation. The second applies in particular to connected or discrete amenable groups, and uses data derived from the irreducible representations appearing in a decomposition over a Cartan subalgebra, and Connes' ideas of transverse measures and functions in the recovery process. Non-trivial examples are also given.