Projections in full \(C^*\)-algebras of semisimple Lie groups
type: Item
instance of: scholarly article
title: Projections in full \(C^*\)-algebras of semisimple Lie groups
zbMATH Open document ID: 0753.22002
DOI: 10.1007/BF01934326
author: Alain Valette
published in: Mathematische Annalen
publication date: 1993-01-12T00:00:00Z
full work available at URL: https://eudml.org/doc/165002
review text: Let \(G\) be a locally compact group. Denote by \(C^*(G)\) the full \(C^*\)-algebra of \(G\), and by \(\mathcal K\) the algebra of compact operators on the infinite-dimensional separable Hilbert space. The main result of the paper runs as follows: let \(G\) be a semisimple Lie group with finite centre; there is no nonzero projection in \(C^*(G)\otimes{\mathcal K}\) if and only if the adjoint group of \(G\) has at least one simple factor isomorphic to an odd-dimensional Lorentz group \(SO_ 0(2n+1,1)\). If \(G\) is almost simple, this is still equivalent to the vanishing of \(K_ 0(C^*(G))\). These results are proved after a preliminary study of minimal projections in \(L^ 1(G)\) and in \(C^*(G)\) (in connection with integrable representations and with Kazhdan's property (T)).
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 19K35
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 86194
zbMATH Keywords: locally compact group
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: Lorentz group
zbMATH Keywords: minimal projections
zbMATH Keywords: integrable representations
zbMATH Keywords: Kazhdan's property (T)
MaRDI profile type: Publication
Wikidata QID: Q115393044
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description: scientific article; zbMATH DE number 86194