The noncommutative topological factor theorem for rank-one product lattices

Abstract

We prove a noncommutative topological factor theorem for irreducible lattices in products of real rank-one simple Lie groups. The intermediate C*-subalgebras between the reduced group C*-algebra and the boundary crossed product are exactly the crossed products arising from coordinate subproducts of the Furstenberg boundary. This follows from a more general theorem for product boundary actions, which also yields tree and mixed local-field versions. We finally show that the corresponding classification for the full flag action of $\operatorname{SL}_3(\mathbb Z)$ would imply ordinary ITAP.

Disclosure

“The main operator-algebraic input is Suzuki’s slice-map theorem [Su17, Proposition 3.4]. For approximation properties we use [CH89, HK94, Sz91]. The dynamical argument uses Poincaré recurrence and Powers’ method [Po75]. AI statement. ChatGPT 5.6 Sol was used at an exploratory stage to in- vestigate the extension of the topological factor theorem to reduced crossed products and the relation between its scalar-expectation case and ITAP. It was subsequently used to draft parts of”

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2026-08-23T05:00:12.241772Z

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