\(K\)-theory of Furstenberg transformation group \(C^{\ast}\)-algebras (Q2862944)

This article computes the \(K\)-theory of crossed product \(C^{\ast}\)-algebras of tori of arbitrary dimension by certain minimal homeomorphisms, called Furstenberg transformations. The basis of this computation is the Pimsner-Voiculescu exact sequence. Since the \(K\)-theory of tori gets more and more complicated with increasing dimension, a full computation of the \(K\)-theory of the crossed product leads to some intricate combinatorics and linear algebra.NEWLINENEWLINECertain Furstenberg transformations are minimal and uniquely ergodic, so that the crossed product \(C^{\ast}\)-algebra is simple with a unique trace. A theorem by Lin and Phillips shows that such crossed products are classified up to isomorphism by \(K\)-theory together with the order on \(K_0\) and the class of the unit; the order on \(K_0\) contains the same information as the map on \(K_0\) induced by the unique trace.NEWLINENEWLINEThe Anzai transformations are a particular case of Furstenberg transformations. They are defined on the \(n\)-torus by mapping NEWLINE\[NEWLINE (z_1,\dots,z_n)\mapsto (\exp(2\pi i\theta) z_1, z_1z_2, z_2z_3,\dots, z_{n-1}z_n) NEWLINE\]NEWLINE for a real number \(\theta\). By homotopy invariance, the \(K\)-theory of the crossed product, as an abelian group, does not depend on \(\theta\); but the order structure depends on \(\theta\). Both \(K_0\) and \(K_1\) are finitely generated abelian groups of the same rank. This common rank depends on the dimension \(n\) of the torus. The resulting sequence is described in several different ways, using generating functions and an interpretation by certain kinds of partitions. This leads to an asymptotic formula for these ranks.