2016/12/13 by Kaliszewski, S., Omland, Tron, Quigg, John
#46L55 (Primary) #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1612.04088
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 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 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.