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تسجيل مستخدم جديدRigidity theory for $C^*$-dynamical systems and the Pedersen Rigidity Problem
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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 Landstads 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 $G$.
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This is a follow-up to a paper with the same title and by the same authors. In that paper, all groups were assumed to be abelian, and we are now aiming to generalize the results to nonabelian groups. The motivating point is Pedersens theorem, which
does hold for an arbitrary locally compact group $G$, saying that two actions $(A,alpha)$ and $(B,beta)$ of $G$ are outer conjugate if and only if the dual coactions $(Artimes_{alpha}G,widehatalpha)$ and $(Brtimes_{beta}G,widehatbeta)$ of $G$ are conjugate via an isomorphism that maps the image of $A$ onto the image of $B$ (inside the multiplier algebras of the respective crossed products). We do not know of any examples of a pair of non-outer-conjugate actions such that their dual coactions are conjugate, and our interest is therefore exploring the necessity of latter condition involving the images, and we have decided to use the term Pedersen rigid for cases where this condition is indeed redundant. There is also a related problem, concerning the possibility of a so-called equivariant coaction having a unique generalized fixed-point algebra, that we call fixed-point rigidity. In particular, if the dual coaction of an action is fixed-point rigid, then the action itself is Pedersen rigid, and no example of non-fixed-point-rigid coaction is known.
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