# Rigidity theory for $C^*$-dynamical systems and the "Pedersen Rigidity Problem", II

## TL;DR

This paper extends rigidity results for $C^*$-dynamical systems from abelian to nonabelian groups, exploring conditions under which dual coactions determine the original actions uniquely, and introduces the concepts of Pedersen and fixed-point rigidity.

## Contribution

It generalizes Pedersen rigidity results to nonabelian groups and investigates the necessity of conditions for dual coactions to determine actions uniquely.

## Key findings

- No known examples of non-outer-conjugate actions with conjugate dual coactions.
- Introduces the concept of Pedersen rigidity for cases where the dual coaction condition is redundant.
- Links fixed-point rigidity of dual coactions to Pedersen rigidity of actions.

## Abstract

This is a follow-up to a paper with the same title and by the same authors. In that paper, all groups were assumed to be abelian, and we are now aiming to generalize the results to nonabelian groups. The motivating point is Pedersen's theorem, which does hold for an arbitrary locally compact group $G$, saying that two actions $(A,\alpha)$ and $(B,\beta)$ of $G$ are outer conjugate if and only if the dual coactions $(A\rtimes_{\alpha}G,\widehat\alpha)$ and $(B\rtimes_{\beta}G,\widehat\beta)$ of $G$ are conjugate via an isomorphism that maps the image of $A$ onto the image of $B$ (inside the multiplier algebras of the respective crossed products). We do not know of any examples of a pair of non-outer-conjugate actions such that their dual coactions are conjugate, and our interest is therefore exploring the necessity of latter condition involving the images, and we have decided to use the term "Pedersen rigid" for cases where this condition is indeed redundant. There is also a related problem, concerning the possibility of a so-called equivariant coaction having a unique generalized fixed-point algebra, that we call "fixed-point rigidity". In particular, if the dual coaction of an action is fixed-point rigid, then the action itself is Pedersen rigid, and no example of non-fixed-point-rigid coaction is known.