Unitary Representations of Lie Groups and Operators of Finite Rank

Abstract

Besides being a principal object of study in harmonic analysis, the convolution algebra of Ll-functions on a locally compact group is traditionally used for investigating (unitary) group representations by methods of associative algebra. With the development of the theory of C*-algebras this role was more and more taken over by the group C*-algebras. A considerable amount of information concerning group C*-algebras, in particular of Lie groups, has accumulated. This article may be regarded as part of a program to use the obtained insights and methods in order to derive results in terms of more classical objects, as Ll-functions or, in the context of Lie groups, even smooth functions, briefly, to prove regularity theorems. To make this more concrete, let me describe two such regularity questions not to be treated in this paper. The topology on the unitary dual G of a given locally compact group G can be characterized as follows. A set A in G is closed if and only if for each ir E G -.. A there exists an f E C* (G) such that 7r(f) =, 0 and p(f) = 0 for all p E A. One may ask whether one can always find an f E L1 (G) with these properties. It turns out that for some groups this is possible, for instance for groups with polynomially growing Haar measure; for others it is not, for instance for noncompact semisimple Lie groups. More information on this topic can be found in [2]. Secondly, as Pukanszky has shown, [40], each primitive quotient C* (G)/P of the group C*-algebra of a connected Lie group G has a unique faithful trace trp. A priori it is not clear that there is an Ll-function f on G such that 0 < trp(f) < oo. But Charbonnel has proved [5], [6], that one can find such an f even in D(G) = Cc (G). In this article we shall solve the following problem in the affirmative. Let ir be a continuous irreducible unitary representation of a connected Lie group H, and suppose that ir(C*(H)) contains the compact operators on the representation space As; i.e., the norm closure of ir (L1 (H)) contains the compact operators. Is it true that ir (Li(H)) contains an operator of rank one? Actually, we shall do somewhat better. We shall construct a smooth function f on H such that ir(f) is an operator of rank one and such that f r