K-theory of Rotation Algebra Crossed Products by Amalgamated Products of Finite Cyclic Groups

2018/09/25 by Sam Walters, Walters, Sam
Mathematics · #(Primary) 46L80 #(Secondary) 19K14 #13D15 #19L10 #46L40 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1809.09286

openalex publication_date 2018/09/25 · openalex created_date 2018/10/05 · openalex updated_date 2026/07/28

Abstract

The K-groups of the crossed product of the rotation C*-algebra A_θ by free and amalgamated products of the cyclic groups \mathbb Z_n, for n=2,3,4,6, are calculated. The actions here arise from the canonical actions of these groups on the rotation algebra under the flip, cubic, Fourier, and hexic automorphisms, respectively. An interesting feature in this study is that although the inclusion A_θ→ A_θ\rtimes \mathbb Z_n induces injective maps on their K₀-groups, the same is not the case for the inclusions A_θ\rtimes \mathbb Z_d → A_θ\rtimes \mathbb Z_n for 2≤ d < n ≤ 6 and d|n, which we endeavor to calculate. Further, while for free products K₁(A_θ\rtimes [\mathbb Z_m ∗ \mathbb Z_n]) = 0, for amalgamated products K₁(A_θ\rtimes [\mathbb Z_m ∗_\mathbb Z_d \mathbb Z_n]) = \mathbb Z^k is non-vanishing (k=1,2).

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