AH crossed products of Cantor extensions and nonliftable Elliott-invariant automorphisms
Year: 2026
Type: preprint
Abstract: The crossed product of any minimal almost one-to-one extension of a Cantor minimal system on a compact metrizable space is approximately homogeneous. We construct two such extensions of the universal odometer with the same Elliott invariant and radius 1/b for every stabilized projection corner of trace b > 0. Under every affine identification of their trace spaces, a common continuous tracial rank is realized by nonunital hereditary subalgebras with relative radii 4 and 1. This answers the hereditary-subalgebra question posed by Hirshberg and Phillips in 2024 within minimal crossed products. For each integer r ≥ 2 and positive weights δ₁,...,δᵣ, we construct a minimal crossed product with ordered K0 = Q^r in the strict coordinate order. For each projection corner of class (a₁,...,aᵣ), its radius is the maximum of δᵢ/aᵢ over 1 ≤ i ≤ r, and its dimension–rank ratio is twice that value. Coordinate permutations act on its full Elliott invariant; when the weights are pairwise distinct, only the identity lifts. The two-state case answers a 2019 question of Hirshberg and Phillips. We compute relative radii at all continuous tracial ranks and characterize bounded nonnegative upper semicontinuous comparison functions. They are closed under nonempty pointwise infima, answering Phillips's 2024 minimum question for these examples. Exact formulas for coordinate-rank jumps include comparison at the critical coefficient. In one odometer example, a single discontinuous rank has relative radii filling [1/2,1] along its affine orbit. Every finite group acts faithfully on the Elliott invariant of this same algebra, fixing K-theory and preserving all projection-corner radii and the collection of continuous comparison functions, while no nonidentity element has a compatible scaled Cuntz-semigroup lift. GitHub repository
Author: Lupini, Martino
Institution: University of Bologna
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225319986