Kazhdan constants and the dual space topology
type: Item
instance of: scholarly article
title: Kazhdan constants and the dual space topology
zbMATH Open document ID: 0787.22006
DOI: 10.1007/BF01444731
author: Eberhard Kaniuth
author: Keith F. Taylor
published in: Mathematische Annalen
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/164972
review text: Let \(G\) be a locally compact group. For a compact subset \(S\) of \(G\) and for \(\pi\in\widehat {G}\) define the Kazhdan constant of \(S\) and \(\pi\) by \[ K_ S(\pi)= \inf_{\xi\in {\mathcal H}_ \pi^ 1} (\max_{x\in S} \|\pi(x)\xi-\xi\|). \] In this paper the authors study the continuity of the function \(\pi\to K_ S(\pi)\). They prove for an almost connected \(G\) that \(K_ S\) is continuous on the whole space \(\widehat {G}\) for some compact subset \(S\) which generates \(G\) if and only if \(G\) contains a compact normal subgroup \(K\) such that \(G/K\) is abelian. Let now \(G\) be a separable locally compact group and \(H\) a closed subgroup of \(G\) such that \(G/H\) is compact. Suppose that \(\Sigma\) is a set of representations of \(H\) which act in the same finite dimensional Hilbert space \({\mathcal H}\). Then \(\sigma\to K_ S(\text{ind}_ H^ G \sigma)\) is a continuous function on \(\Sigma\). The authors apply this result to Moore groups and to motion groups where they show that \(K_ S\) is continuous on large open subsets of \(\widehat {G}\). Finally they show for acceptable connected semi-simple groups, that \(K_ S\) is continuous on the set of irreducible principal series representations.
reviewed by: Jean Ludwig
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 22D15
Mathematics Subject Classification ID: 22D30
zbMATH DE Number: 126347
zbMATH Keywords: dual space
zbMATH Keywords: locally compact group
zbMATH Keywords: Kazhdan constant
zbMATH Keywords: Moore groups
zbMATH Keywords: motion groups
MaRDI profile type: Publication
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OpenAlex ID: W2024851093
description: scientific article; zbMATH DE number 126347