Dynamical Perturbing and $C^*$-algebra Lifting Problems
Samantha Pilgrim
TL;DR
This work initiates and develops a theory of almost-actions: sequences of near-morphisms from a group to homeomorphisms of a compact space that are asymptotically multiplicative. It establishes topological stability results in cases such as finite $\Gamma$ acting on Cantor sets, yielding genuine actions near almost-actions and finite approximations for actions of virtually free groups on Cantor sets, with consequences for residual finiteness and finite-dimensional approximations of crossed products. The authors introduce a Cartan-pair lifting framework and a notion of conditional semiprojectivity to translate dynamical stability questions into $C^*$-algebraic lifting problems, proving finite-dimensional algebras are cond. semiprojective but certain inclusions are not. They construct representations from almost-actions, relate these to Cartan structures, and provide a general abstract perturbing-problem formulation for finite groups, offering conjectures and directions to further connect dynamical stability with operator-algebraic lifting theory and finite-dimensional approximations of crossed products.
Abstract
Approximate morphisms have seen significant study across many areas of mathematics, for instance, in the theory of Absolute (Neighborhood) Retracts in topology, or of almost-commuting unitary matrices in analysis. This paper initiates study of a type of approximate group action (which we call almost-actions). More precisely, these are sequences of set maps from a group into the homeomorphisms of a compact metric space which are asymptotically multiplicative in the sense of the metric. We prove a kind of topological stability holds in certain cases, such as when the group is finite and the space is a Cantor set, so that one can find genuine actions near the almost-actions, and apply these results to produce new finite approximations of many actions by virtually free groups on Cantor sets. We also introduce a new type of lifting problem for $C^*$-algebras which, rather than asking for a lift of a homomorphism, asks for a lift of the structure of a Cartan pair, and use this new notion to characterize the stability of more general almost-actions. In the course of attempting to apply the theory of semiprojective $C^*$-algebras to these questions, we define a notion of conditional semiprojectivity for morphisms of $C^*$-algebras. We show that maps of finite-dimensional $C^*$-algebras are conditionally semiprojective, but that the inclusion of $C(S^1)$ into $C(S^1)\rtimes Γ$ (for any non-trivial action of a finite group $Γ$) is not. We conclude with a conjecture about the general stability of almost-actions by finite groups and some commentary on possible directions for further developing these ideas.