Unitary representations of Lie groups and operators of finite rank
type: Item
instance of: scholarly article
title: Unitary representations of Lie groups and operators of finite rank
zbMATH Open document ID: 0828.22013
DOI: 10.2307/2118617
published in: Annals of Mathematics. Second Series
publication date: 1996-01-02T00:00:00Z
full work available at URL: https://pub.uni-bielefeld.de/record/1775498
review text: Let \(H\) be a connected Lie group, let \(C^*(H)\) be its \(C^*\)-algebra and let \(\pi \in \widehat {H}\) be an irreducible representation of \(H\) such that \(\pi (C^*(H))\) contains the compact operators. In the semi- simple case, there exist smooth functions \(f\) with compact support on \(H\) such that \(\pi(f)\) is a rank one projection. This fact is due to the bounded ``finiteness'' of the \(K\)-types. In the nilpotent case, Dixmier's functional calculus gives us Schwartz-functions \(f\) such that \(\pi (f)\) is of finite rank. In this paper the author treats the general case and he shows that there exists for any \(H\) and \(\pi\) as above a smooth function \(f\) on \(H\), such that \(r \cdot f\) is integrable for any representative function \(r\) of \(H\) and such that \(\pi(f)\) is an operator of rank 1. Representative functions are matrix coefficients of finite dimensional representations of \(H\). These functions play an essential role in the proof. In the first part of the paper the author discusses representative functions on groups, he introduces a special class of Lie groups, the class \([MB]\), which are semidirect products of abelian groups \(B\) and a group \(M\), whose connected component \(M_0\) allows a locally faithful finite dimensional representation. In sections 3 to 7 the (long and difficult) proof of the theorem for groups of type \([MB]\) is given. It proceeds by induction on the dimension of \(M_0\) and considers the different possibilities for the restriction of \(\pi\) to the nilradical of \(H\). The techniques used in these proofs are too involved to be explained here. Finally in section 8 it is shown that any connected simply connected Lie group \(H\) can be imbedded into a larger \([BM]\) group \(G\) and \(\pi\) can be extended to a representation \(\rho\) of \(G\) in such a way that the existence of a rapidly decreasing \(f \in L^1(G)\) with \(\text{rank}(\rho (f)) < \infty\) implies the existence of a \(g \in L^1(H)\) such that \…
Mathematics Subject Classification ID: 22E27
zbMATH DE Number: 750881
zbMATH Keywords: connected Lie groups
zbMATH Keywords: operators of finite rank in \(\pi(C^* (G))\)
zbMATH Keywords: \(L^ 1(G)\)
zbMATH Keywords: irreducible representation
Wikidata QID: Q115229221
author: D. Poguntke
reviewed by: Jean Ludwig
MaRDI profile type: Publication
OpenAlex ID: W2011081759
description: scientific article; zbMATH DE number 750881