Dynamical Perturbing and $C^*$-algebra Lifting Problems

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 $C(S^1)\rtimes \Gamma$ (for any non-trivial action of a finite group $\Gamma$) 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.