Group extensions and Plancherel formulas (Q1080535)

Let G be a separable locally compact group. The point measure at e defines, if G is unimodular, a trace on \(C^*_+(G)\), and a semitrace if G is not unimodular. The author regards the Plancherel formula for G as being the decomposition of this semitrace into semicharacters. These notions were first systematically studied by \textit{N. V. Pedersen} [Math. Ann. 247, 191-244 (1980; Zbl 0406.22008)]. The first section of the paper collects some facts about induction and direct integral decompositions of semitraces and projective semitraces. The next section contains the abstract Plancherel formula. Then the author proceeds to give the explicit Plancherel formula for a number of interesting groups, including the Mautner group [\textit{M. Cowling}, Boll. Unione Mat. Ital., V. Ser., A 15, 616-623 (1978; Zbl 0392.43010) has also produced a Plancherel formula for this group], a 7-dimensional group of Dixmier, and some Heisenberg- type groups in which the role of \({\mathbb{R}}^ n\) is played by a separable locally compact abelian group. In the next section, the author describes the Plancherel formula for G in terms of the Plancherel formula for a normal subgroup N. The situation he treats is quite general, so that, for example, he must assume that almost all the stability groups are closed and that for almost all \(\pi\in \hat N\), the orbit \(G_{\pi}\) is canonical. However, the main assumption he makes is that the stability groups are locally constant. That is, if \(\mu^ N=\int_{X}\mu_ t dv(t)\) is the decomposition of the Plancherel measure \(\mu^ N\) of N into measures ergodic under the action of G, then for almost all t, there is a G-invariant Borel set \(E_ t\) supporting \(\mu_ t\) on which the stability groups are constant. This replaces the assumption that N is almost regularly embedded in G. With these assumptions, the author gives the Plancherel formula for G in terms of the Plancherel formulas for N, the stability groups and the action of G on them. With the assumption that N has a type I regular representation the objects that appear in the Plancherel formula are given more explicitly.