Projective Limits in Harmonic Analysis

Abstract

phisms, functors are produced which on objects are given by G -* L (G),Ll(G), M(G), W(G),denoting, resp., the L -space,L -algebra, measure algebra, and von Neu-1 2 mann algebra generated by left regular representation of L on L . All functors but but the second are shown to preserve projective limits; by example, the second is shown not to do so. The category of Hubert spaces and linear transformations of norm < 1 is shown to have projective limits; some propositions on such limits are given. Also given is a type and factor characterization of projective limits in the category of W* -algebras and surjective normal *-algebra homomorphisms.0. Introduction. In the theory of topological groups, two well-known results are the following: Every compact group is the projective limit of closed subgroups of unitary groups U(n) (n < °°). Every locally compact group contains an open subgroup which is the projective limit of Lie groups. These results indicate why projective limits are of interest-loosely speaking, one has made a reduction to a simpler case. In this paper we investigate whether certain mathematical objects studied by harmonic and functional analysts have projective limit decompositions which reflect decompositions presumed to exist on the group level.In § 1 we present a treatment of induced transformations of measures and measurable functions. In particular, given tp: G -► H in the category of locally compact groups and continuous, proper (defined within) surjective group homomorphisms, we produce three functors which to G assign, resp., the objects L2(G), M(G), and Ll(G), and which to <ï> assign morphisms appropriate to these objects. In §2 we show that projective (and injective) limits exist in the category of Hubert spaces and norm-decreasing linear transformations, and present some propositions on such limits. In §3 a fourth functor is introduced, which to the

The spine of a Fourier-Stieltjes algebra

“…However, it is not clear that G S is locally compact, in general. The following special case seems known (there is such a comment on page 133 of [20], for example). Since we could not find a proof in any of the standard references [22], [24] or [31], we will supply a brief one.…”