The Pedersen Rigidity problem

If α is an action of a locally compact abelian group G on a C^∗-algebra A, Takesaki-Takai duality recovers (A, α) up to Morita equivalence from the dual action of Ĝ on the crossed product A ⋊ _αG. Given a bit more information, Landstad duality recovers (A, α) up to isomorphism. In between these, by modifying a theorem of Pedersen, (A, α) is recovered up to outer conjugacy from the dual action and the position of A in M(A ⋊ _αG). Our search (still unsuccessful, somehow irritating) for examples showing the necessity of this latter condition has led us to formulate the "Pedersen Rigidity problem". We present numerous situations where the condition is redundant, including G discrete or A stable or commutative. The most interesting of these "no-go theorems" is for locally unitary actions on continuous-trace algebras.