Simple infinite-dimensional quotients of the group \(C^{\bigstar}\)-algebras of certain discrete 6-dimensional nilpotent groups
type: Item
instance of: scholarly article
title: Simple infinite-dimensional quotients of the group \(C^{\bigstar}\)-algebras of certain discrete 6-dimensional nilpotent groups
zbMATH Open document ID: 1071.22003
DOI: 10.1016/S0019-3577(05)80016-3
author: Kamran Reihani
author: Ali Reza Medghalchi
published in: Indagationes Mathematicae. New Series
publication date: 2005-06-10T00:00:00Z
review text: There is a natural one-to-one correspondence between the dual \(\hat G\), i.e. the set of unitary equivalence classes of irreducible unitary representations of a locally compact group \(G\), and the unitary equivalence classes of irreducible *-representations of the corresponding group \(C^*\)-algebra \(C^*(G)\). To each representation \(\pi\in \hat G\) one associates its kernel \(I_\pi \triangleleft C^*(G)\), one obtains a map \(\hat G \to Prim(G)\) from the dual \(\hat G\) to the space \(Prim(G)\) of primitive ideals in \(C^*(G)\). The quotient \(C^*(G)/I_\pi\) becomes a simple quotient if the primitive ideal \(I_\pi\) is maximal. In the paper the authors study the simple quotients of group \(C^*\)-algebras of two discrete 6-dimensional nilpotent groups \(H_{6,1}\) and \(H_{6,2}\) from the list of 24 6-dimensional nilpotent groups \(H_{6,j}\), \(j=\overline{1,24}\), as the higher-dimensional analogues of the irrational rotation algebras. The main results are: (1) (Theorem 2.1) the faithful simple quotient \(A^{6,1}_\theta\) of \(C^*(H_{6,1})\) is presented as some \(C^*\)-crossed product \(A^{6,1}_\theta \cong \mathcal C(\mathbb T^3) \rtimes_\Phi \mathbb Z^2\) and (Theorem 3.1) \(A^{6,2}_\theta \cong \mathbb C(\mathbb T^3) \rtimes_\Psi \mathbb Z^2\); (2) a matrix presentation for the non-faithful simple quotients (Theorems 2.4, 2.8, 3.4, 3.8). The authors compute also the corresponding \(K\)-groups.
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 19K14
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 2175010
zbMATH Keywords: nilpotent group
zbMATH Keywords: group C*-algebra
zbMATH Keywords: simple quotient
zbMATH Keywords: C*-crossed product
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2175010