Finite group actions on AF algebras obtained by folding the interval
type: Item
instance of: scholarly article
title: Finite group actions on AF algebras obtained by folding the interval
zbMATH Open document ID: 0821.46088
DOI: 10.1007/BF00961400
author: Ola Bratteli
author: David E. Evans
author: Akitaka Kishimoto
author: George A. Elliott
published in: \(K\)-Theory
publication date: 1995-09-28T00:00:00Z
review text: The paper extends previous work by \textit{B. Blackadar} and \textit{A. Kumjian} [cf. Ann. Math., II. Ser. 131, No. 3, 589-623 (1990; Zbl 0718.46024) and C. R. Math. Acad. Sci., Soc. R. Can. 10, No. 5, 217-218 (1988; Zbl 0669.46028), respectively] who constructed AF algebras allowing an automorphism of order 2, the corresponding fixed point algebra being not AF. The present approach starts from algebras whose spectrum is an interval with multiple end points and with morphisms coming from folding and cutting the interval. The inductive limit of these algebras is an AF algebra \(\mathcal A\), and if \(d\) is the number of points at each end of the interval it allows an action \(\alpha\) of a finite group \(G\) of order \(d> 1\). The dimension group of \(\mathcal A\) is \(\mathbb{Z}\left[{1\over d}\right]\oplus \mathbb{Z}^{d- 1}\) and the fixed point algebra \({\mathcal A}^ \alpha\) has \(K\)-groups \(K_ 0({\mathcal A}^ \alpha)= \mathbb{Z}\left[{1\over d}\right]\) and \(K_ 1({\mathcal A}^ \alpha)= \mathbb{Z}^{d'- 1}\), where \(d'\) is the order of the dual of \(G\). In particular, \({\mathcal A}^ \alpha\) is not AF. A modification of the construction shows that \(\mathcal A\) can be chosen UHF.
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 19K14
Mathematics Subject Classification ID: 46L35
zbMATH DE Number: 720847
zbMATH Keywords: AF algebras
zbMATH Keywords: automorphism of order 2
zbMATH Keywords: fixed point algebra
zbMATH Keywords: \(K\)-groups
MaRDI profile type: Publication
cites work: Symmetries of the CAR algebra
cites work: Q3730469
cites work: Inductive Limits of Finite Dimensional C ∗ -Algebras
cites work: Q3822695
full work available at URL: https://doi.org/10.1007/bf00961400
OpenAlex ID: W2100164771
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description: scientific article; zbMATH DE number 720847