TY - JOUR

T1 - Rigidity theory for C∗-dynamical systems and the "pedersen Rigidity Problem", II

AU - Kaliszewski, Steven

AU - Omland, Tron

AU - Quigg, John

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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 Pedersen's theorem, which does hold for an arbitrary locally compact group G, saying that two actions (A,α) and (B,β) of G are outer conjugate if and only if the dual coactions (A αG,α) and (BβG,β) 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.

AB - 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 Pedersen's theorem, which does hold for an arbitrary locally compact group G, saying that two actions (A,α) and (B,β) of G are outer conjugate if and only if the dual coactions (A αG,α) and (BβG,β) 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.

KW - Crossed-product

KW - exterior equivalence

KW - generalized fixed-point algebra

KW - outer conjugacy

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U2 - 10.1142/S0129167X19500381

DO - 10.1142/S0129167X19500381

M3 - Article

AN - SCOPUS:85068193007

JO - International Journal of Mathematics

JF - International Journal of Mathematics

SN - 0129-167X

ER -