Recursive subhomogeneity of orbit-breaking subalgebras of $\mathrm{C}^*$-algebras associated to minimal homeomorphisms twisted by line bundles
Marzieh Forough, Ja A Jeong, Karen R. Strung
TL;DR
This work establishes an explicit recursive subhomogeneous decomposition for the Cuntz–Pimsner algebra O(𝔈_Y) associated to orbit-breaking bimodules twisted by a line bundle over a minimal dynamical system. The construction proceeds by embedding O(𝔈_Y) into a direct sum of endomorphism bundles Γ(𝔐_k), then describing the image via a boundary-decomposition property and assembling an iterated pullback diagram of homogeneous pieces Γ(𝔐_k). The main theorem yields an RSH decomposition with length at least K, where K is the number of distinct first-return times to the breaking set Y, and the base spaces have dimension controlled by X while the matrix sizes are r_1, …, r_K. These results enable, in particular, Z-stability and classification consequences mirroring the untwisted case and provide a robust framework for analyzing CP algebras twisted by line bundles in orbit-breaking settings. The construction generalizes prior orbit-breaking decompositions for crossed products and sets the stage for subsequent K-theory and stable rank analyses in twisted dynamical C*-algebras.
Abstract
In this paper, we construct a recursive subhomogeneous decomposition for the Cuntz--Pimsner algebras obtained from breaking the orbit of a minimal Hilbert $C(X)$-bimodule at a subset $Y \subset X$ with non-empty interior. This generalizes the known recursive subhomogeneous decomposition for orbit-breaking subalgebras of crossed products by minimal homeomorphisms.