The elements of bounded trace in the \(C^*\)-algebra of a nilpotent Lie group
type: Item
instance of: scholarly article
title: The elements of bounded trace in the \(C^*\)-algebra of a nilpotent Lie group
zbMATH Open document ID: 0587.22003
DOI: 10.1007/BF01388757
author: Jean Ludwig
author: Günther Rosenbaum
author: Jürgen Samuel
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143308
review text: Let G be a connected, simply connected nilpotent Lie group with Lie algebra \({\mathfrak g}\). We identify the dual space \(\hat G\) of G with the space \({\mathfrak g}^*/Ad^*(G)\) of the G-orbits in \({\mathfrak g}^*\) under the coadjoint action. For an integer d, denote by \(\hat G_ d\) (resp. by \(\hat G_{<d})\) the subset of \(\hat G,\) consisting of the orbits of dimension d (resp. of the orbits of dimension strictly less than d). In this paper we study the following question. Let A be a closed subset of \(\hat G.\) What positive elements x in the \(C^*\)-algebra \(C^*(G)\) of G are with bounded trace on A, i.e., for what \(x\geq 0\) in \(C^*(G)\), is sup \(\{\) tr \(\pi\) (x) \(|\) \(\pi \in A\}<\infty ?\) It turns out, that the subsets \(C_ d:=(A\cap \hat G_ d)^-\cap \hat G_{<d}\) (where \({}^-\) means closure in \(\hat G)\), \(d=2,4,6,...\), play a crucial role. Let \(C=\cup_{d}C_ d\). In fact, it is easily shown, that a necessary condition for x to be with bounded trace on A, is that \(\pi (x)=0\) for every \(\pi\in C\) (Theorem 3.4.). Let ker C:\(=\{\cap\) ker \(\pi\) \(| \pi \in C\}\). The main result of the paper is Theorem 5.3.: Every \(0\leq x\) in the Pedersen ideal j(C) of ker C is with bounded trace on A. In the proof of this theorem we use the method of variable groups. This allows us to proceed inductively on the dimension of G and to use Kirillov's lemma at every induction step. In the proof we also need the following results. For any closed subset B of \(\hat G,\) the Pedersen ideal j(B) in \(C^*(G)\) is generated by Schwartz functions (Theorem 2.7.). Let \(V=\{v_ 1,...,v_ n\}\) be a Jordan-Hölder basis of \({\mathfrak g}^*\) and let for \(l\in {\mathfrak g}^*\), \(P(z,l)=\sum^{n}_{j=1}P_ j(z,l)\nu_ j\) be the canonical polynomial map, describing the G-orbit of l \((z\in {\mathbb{R}}^ d\), \(d=\dim G\cdot l)\). Let \(1\leq e_ 1<e_ 2<...<e_ d\leq n\) and denote by R the polynomial mapping \(R(z):=(P_{e_ 1}(z,l),...,P_{e_ d}(z,l))\in {\mathbb{R}}^…
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 3941812
zbMATH Keywords: dual of nilpotent Lie group
zbMATH Keywords: Kirillov induction
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: bounded trace
zbMATH Keywords: Pedersen ideal
zbMATH Keywords: variable groups
zbMATH Keywords: Schwartz functions
MaRDI profile type: Publication
Wikidata QID: Q115393639
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description: scientific article; zbMATH DE number 3941812