An Extension of Plancherel's Formula to Separable Unimodular Groups

Abstract

We show that for any f square-integrable on a separable unimodular locally compact group G (relative to Haar measure), the integral over G of the square of the absolute value of f equals the integral over a certain measure space of the square of the (at each point of the space) of the Fourier transform of f. By relative norm we mean that defined by von Neumann [5] for factors, and for the present we define the Fourier transform thru the use of the von Neumann reduction theory [6]. The formula which is obtained here has as instances, though not as direct corollaries, the formula of Plancherel, its generalization to separable locally compact abelian groups, the Peter-Weyl theorem, and a formula recently obtained by Gelfand and Neumark [2] for the case of the Lorentz group. The measure space in question has its measure ring isomorphic to the Boolean ring B of closed linear manifolds in L2(G) invariant under both left and right translations, and the corresponding measure is uniquely determined, modulo normalization of the at each point of the space. This measure ring acts as a kind of measure-theoretic to the group. For example, making the so-called standard normalization of the norm, if G is abelian, then B as a measure ring is abstractly identical with the measure ring of the character group of G, under Haar measure; if G is compact, then B is abstractly the measure ring of a discrete set of points, the set being in one-to-one correspondence with the collection of equivalence classes of continuous irreducible representations of G, the measure of a point being proportional to the degree of the corresponding representation. Basic in our work is a certain countably-additive non-negative function, which we call the dual of G, defined on the lattice of all projections in the weak closure V of the algebra generated by left translations in L2(G),-or alternatively, on the lattice of closed linear manifolds in L2(G) invariant under right translations. The gage, which is defined in an intrinsic fashion, is invariant under transformation by unitary operators in V, and is in fact a weight function in the terminology of von Neumann [6]. His reduction theory consequently yields a representation of the gage as an integral over the measure space described above of constituents (relative dimension functions) arising from factors. This representation together with the facts that the convolution operator on L2(G) defined by a self-adjoint element of L2(G) is hypermaximal symmetric (Ambrose [1]) and that every bounded linear operator on L2(G) which commutes with all right translations is in V (Segal [4]), are the principal known results used in our derivation of the generalized Plancherel formula.