Journal of Operator Theory

Volume 95, Issue 1, Winter 2026 pp. 159-188.

Dynamical propagation and Roe algebras of warped space

Authors: Tim de Laat (1), Federico Vigolo (2), Jeroen Winkel (3)
Author institution: (1) Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
(2) Mathematisches Institut, Georg-August Universitaet Goettingen, 37073 Goettingen, Germany
(3) Mathematical Institute, University of Muenster, 48149 Muenster, Germany

Summary: Given a non-singular action $\Gamma \curvearrowright (X,\mu)$, we define the $*$-algebra $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ of operators of finite dynamical propagation associated with this action. This assignment is completely canonical and depends only on the class of measures of $\mu$. We prove that the algebraic crossed product $L^{\infty}X {\rtimes_{\rm alg}} \Gamma$ surjects onto $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ and that this surjection is a $\ast$-isomorphism whenever the action is essentially free. As a consequence, we canonically characterize ergodicity and strong ergodicity of the action in terms of structural properties of $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ and its closure. We also use these techniques to describe the Roe algebra of a warped space in terms of the Roe algebra of the (non-warped) space and the group action. We apply this result to Roe algebras of warped cones.

DOI: http://dx.doi.org/10.7900/jot.2024apr09.2486
Keywords: Roe algebras, finite (dynamical) propagation, strong ergodicity, warped metric, crossed products

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