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

TL;DR

This paper explores rigidity in $C^*$-dynamical systems, focusing on conditions under which actions are outer conjugate and when the images of algebras are preserved, with particular results for discrete groups.

Contribution

It provides new insights into the Pedersen rigidity problem, identifying situations where the preservation of algebra images is unnecessary and showing the impossibility of distinct fixed-point algebras in certain cases.

Findings

For discrete groups, the condition on algebra images is always redundant.

In some cases, having distinct generalized fixed-point algebras is impossible.

Several situations are identified where the rigidity condition can be omitted.

Abstract

Let $G$ be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of $G$ on $C^*$-algebras $A$ and $B$ are outer conjugate if and only if there is an isomorphism of the crossed products that is equivariant for the dual actions and preserves the images of $A$ and $B$ in the multiplier algebras of the crossed products. The rigidity problem discussed in this paper deals with the necessity of the last condition concerning the images of $A$ and $B$. There is an alternative formulation of the problem: an action of the dual group $\hat G$ together with a suitably equivariant unitary homomorphism of $G$ give rise to a generalized fixed-point algebra via Landstad's theorem, and a problem related to the above is to produce an action of $\hat G$ and two such equivariant unitary homomorphisms of $G$ that give distinct generalized fixed-point algebras. We present several situations where the condition on the images of $A$ and $B$ is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if $G$ is discrete, this will be the case for all actions of $G$.