Finite group actions on AF algebras obtained by folding the interval
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.
scientific article; zbMATH DE number 720847
scientific article