Bounded trace \(C^*\)-algebras and integrable actions
type: Item
instance of: scholarly article
title: Bounded trace \(C^*\)-algebras and integrable actions
zbMATH Open document ID: 1083.46037
author: Klaus Deicke
author: Robert J. Archbold
published in: Mathematische Zeitschrift
publication date: 2005-06-02T00:00:00Z
review text: Section 1 of this paper relates the ``strength of convergence'' in orbit spaces to the properties of transformation groups (actions) and their \(C^*\)-algebras. By Green's theorem, given a free action of an l.c.\ group \(G\) on a l.c.\ Hausdorff space \(X\), the crossed product \(C_0(X)\times G\mathrel{=:}{\mathcal C}\) has continuous trace iff \(G\) acts properly. Then the upper multiplicity \(M_{U}\) of irreducible representations of \({\mathcal C}\) cannot exceed 1. For a more general integrable action introduced by M.~A.\ Rieffel, A.~an Huef proved that \({\mathcal C}\) has bounded trace (BT). A \(C^*\)-algebra \({\mathcal A}\) has BT if and only if \(M_{U}(\pi )<\infty\) for any \(\pi \in\hat A\), see \textit{R.~J.\ Archbold, D.~W.~B.\ Somerset} and \textit{J.~S.\ Spielberg} [J. Funct. Anal. 146, 430--463 (1997; Zbl 0909.46049)]. The paper under review treats similar problems in the setting of (noncommutative) \(C^*\)-dynamical systems \((A,G,\alpha )\) with the arising action of \(G\) on the spectrum \(\hat A\). Section 2 estimates \(M_U\) of a certain representation \(\mathop{\text{Ind}}\pi \) of the crossed product \(A\times {}_\alpha G\) induced from a representation \(\pi \) of \(A\). For a separable system (\(A\) is separable, \(G\)-second countable), \(\mathop{\text{Ind}}\pi \) is irreducible for each \(\pi \in\hat A\) if \(A\) is type I, \(G\) acts freely on \(\hat A\) and any orbit \(G(\pi )\) is locally closed (i.e., open in its closure). Then \(M_{U}(\mathop{\text{ Ind}}\pi)\geq k\in{\mathbb N}\) if there is a sequence \(\pi _n\) in \(\hat A\) and \(k\) sequences \(t_{j,n}\) in \(G\), \(1\leq j\leq k\), so that \(t_{j,n}^{-1}\to\pi \) \ \(\forall\,j\), and, for \(i<j\), \ \(t_{j,n}^{-1}t_{i,n}\to\infty\) as \(n\to\infty\). This entails a part of Green's theorem and has other useful corollaries. The principal Section 3 generalizes \textit{A.~an Huef}'s Theorem 4.8 of [Indiana Univ. Math. J. 51, 1197--1233 (2002; Zbl 1032.46078)], viz., if \((A,G,\alph…
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 2173668
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: irreducible representation
zbMATH Keywords: upper multiplicity
zbMATH Keywords: bounded trace
zbMATH Keywords: \(C^*\)-dynamical system
zbMATH Keywords: \(G\)-space
zbMATH Keywords: action on the spectrum
zbMATH Keywords: (almost) free action
zbMATH Keywords: integrable action
reviewed by: Andrey V. Bulinski
MaRDI profile type: Publication
full work available at URL: s00209-004-0759-4
OpenAlex ID: W2022629860
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DOI: 10.1007/S00209-004-0759-4
description: scientific article; zbMATH DE number 2173668