On the \(K\)-theory of boundary \(C^*\)-algebras of \(\tilde A_2\) groups
type: Item
instance of: scholarly article
zbMATH Open document ID: 1260.46052
author: Oliver King
author: Guyan Robertson
publication date: 2012-08-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1104.4416
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20G25
zbMATH DE Number: 6072571
zbMATH Keywords: affine building
zbMATH Keywords: boundary
zbMATH Keywords: operator algebra
MaRDI profile type: Publication
OpenAlex ID: W1992370611
arXiv ID: 1104.4416
cites work: Groups acting simply transitively on the vertices of a building of type \(\tilde A_ 2\). I
cites work: Property \((T)\) and \(\widetilde{A}_ 2\) groups
cites work: Euler characteristics and Gysin sequences for group actions on boundaries
cites work: Asymptotic \(K\)-theory for groups acting on \(\widetilde A_2\) buildings
cites work: Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras
cites work: C*-Algebras Arising from Group Actions on the Boundary of a Triangle Building
DOI: 10.1017/IS011005004JKT158
published in: Journal of \(K\)-Theory
title: On the \(K\)-theory of boundary \(C^*\)-algebras of \(\tilde A_2\) groups
review text: Let \(\mathbb K\) be a local field with residue field of order \(q\), and let \(\Gamma\) be an \(\widetilde{A_2}\) subgroup of \(\text{PGL}_3(\mathbb K)\). Let \(C(\mathbb P_{\mathbb K}^2,\mathbb Z)\) be the \(\Gamma\)-module of integer-valued continuous functions on the projective plane \(\mathbb P_{\mathbb K}^2\) over \(\mathbb K\).NEWLINENEWLINEIt is shown that the module of coinvariants \(C(\mathbb P_{\mathbb K}^2,\mathbb Z)_\Gamma=H_0(\Gamma;C(\mathbb P_{\mathbb K}^2,\mathbb Z))\) is finite and the class \([\mathbf 1]\) in \(C(\mathbb P_{\mathbb K}^2,\mathbb Z)_\Gamma \) has order bounded by \(q^2-1\). Its order equals \(q-1\) under the assumption that \(\Gamma\) is of Tits type and that \(q\neq 1\;(\text{mod}\,3)\).NEWLINENEWLINELet \(\Omega\) be the Furstenberg boundary of \(\text{PGL}_3(\mathbb K)\) and let \(C(\Omega)\rtimes\Gamma\) be the crossed product \(C^*\)-algebra. It is shown that the order of \([\mathbf 1]_{K_0}\) in the \(K_0\) group of \(C(\Omega)\rtimes\Gamma\) also equals \(q-1\) under the same assumption.
reviewed by: Vladimir M. Manuilov
description: scientific article; zbMATH DE number 6072571