Rokhlin dimension for actions of residually compact groups

We introduce the concept of Rokhlin dimension for actions of residually compact groups on C*-algebras, which extends and unifies previous notions for actions of compact groups, residually finite groups and the reals. We then demonstrate that finite nuclear dimension (respectively, absorption of a strongly self-absorbing C*-algebra) is preserved under the formation of crossed products by residually compact group actions with finite Rokhlin dimension (respectively, finite Rokhlin dimension with commuting towers). Furthermore, if second countable residually compact group contains a non-open cocompact closed subgroup, then crossed products arising from actions with finite Rokhlin dimension are stable. Finally, we study the relationship between the tube dimension of a topological dynamical system and the Rokhlin dimension of the induced C*-dynamical system.

💡 Research Summary

This paper introduces a unified notion of Rokhlin dimension for actions of residually compact groups on C∗‑algebras, thereby extending all previously known versions (compact groups, residually finite groups, and the real line). A residually compact group is a second‑countable locally compact group that admits a sequence σ of closed co‑compact subgroups whose intersection is trivial. Unlike earlier treatments, the authors do not require the approximating subgroups to be decreasing, discrete, or normal; they only assume a “regular” approximation, which includes the decreasing discrete normal case studied by Szabó.

The Rokhlin dimension dim_Rok(α,σ) is defined via equivariant completely positive contractive order‑zero maps φₗ:C(G/H)→A∞∩A′ (ℓ=0,…,d) for each H∈σ, with the sum of the images of the unit equal to 1. The supremum over H gives the overall dimension, and imposing commutation of the ranges yields the Rokhlin dimension with commuting towers. This definition recovers the earlier formulations for finite groups, compact groups, residually finite groups, and ℝ.

The authors then prove several structural preservation results for crossed products A⋊αG when the action has finite Rokhlin dimension.

Theorem B (Nuclear dimension). For a separable C∗‑algebra A, a second‑countable residually compact group G with a regular approximation σ, and a continuous action α, one has
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