Rigidity theory for C∗-dynamical systems and the “Pedersen rigidity problem”

Let (Formula presented.) be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of (Formula presented.) on (Formula presented.)-algebras (Formula presented.) and (Formula presented.) 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 (Formula presented.) and (Formula presented.) 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 (Formula presented.) and (Formula presented.). There is an alternative formulation of the problem: an action of the dual group (Formula presented.) together with a suitably equivariant unitary homomorphism of (Formula presented.) 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 (Formula presented.) and two such equivariant unitary homomorphisms of (Formula presented.) that give distinct generalized fixed-point algebras. We present several situations where the condition on the images of (Formula presented.) and (Formula presented.) is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if (Formula presented.) is discrete, this will be the case for all actions of (Formula presented.).