The Two-Sided Regular Representation of a Unimodular Locally Compact Group

Abstract

We show that for any unimodular locally compact group G, the lattice of all closed linear subspaces in L2(G) which are invariant under right and left translations is a Boolean algebra. (L2(G) is the Hilbert space of functions on G squareintegrable relative to Haar measure.) This Boolean algebra B can be regarded as a kind of dual to the group, being in measure-theoretic respects at least, equivalent to the usual duals, in the cases for which a duality theory is already established (i.e. for abelian and compact groups). In particular, the decomposition of L2(G) relative to B can be looked on as a formal generalization of the Plancherel and Peter-Weyl theorems, in which integration over the character group or over the space of equivalence classes of irreducible representations, is generalized by integration over B. The orthogonality of coordinates of inequivalent irreducible representations has as a formal analog the fact that any two disjoint elements of B are orthogonal. The key result, on which these developments rest, asserts that the only bounded linear transformations on L2(G) which commute with all left translations are the right translations, and the (weak) limits of their linear combinations. In the case of a discrete group this theorem is known [2], but the proof for that case seems incapable of generalization, as it makes heavy use of the special nature of the group. Although the generalized Plancherel and Peter-Weyl theorems can be regarded as stating the decomposition of L2(G) under a one-sided regular representation of G (e.g. under the left regular representation, a -+ La for a e G, where (Laf) (x) = f(a-lx) for f e L2(G)), this viewpoint seems unlikely to lead to a unique analog to those theorems for arbitrary unimodular locally compact groups. The main difficulty which arises may be described briefly as follows. A complete reduction of a unitary representation of a group can be considered to amount to a choice of a maximal Boolean algebra of closed linear subspaces which are invariant under the group. Now in general, this Boolean algebra will not be unique, even within unitary equivalence. For example, in the case of the reduction of an infinite-dimensional Hilbert space under the action of the identity operator, there is one maximal Boolean algebra of (invariant) subspaces which is generated by its minimal elements, and another which contains no minimal elements. In the case of a one-sided regular representation of a group which is abelian or compact, the algebra happens to be unique (within unitary equivalence), for the following reasons: in the abelian case, the invariant subspaces under either onesided regular representation are obviously the same as those invariant under the two-sided (i.e. both) representations; in the compact case, the two-sided regular