Rigidity theory for C ^∗ -dynamical systems and the "pedersen rigidity problem"

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 Ĝ 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 Ĝ 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.