TY - JOUR
T1 - Three versions of categorical crossed-product duality
AU - Kaliszewski, Steven
AU - Omland, Tron
AU - Quigg, John
N1 - Publisher Copyright:
© 2016, University at Albany. All rights reserved.
PY - 2016
Y1 - 2016
N2 - In this partly expository paper we compare three different categories of C*-algebras in which crossed-product duality can be for mulated, both for actions and for coactions of locally compact groups. In these categories, the isomorphisms correspond to C*-algebra isomor phisms, imprimitivity bimodules, and outer conjugacies, respectively. In each case, a variation of the fixed-point functor that arises from classical Landstad duality is used to obtain a quasi-inverse for a crossed product functor. To compare the various cases, we describe in a formal way our view of the fixed-point functor as an “inversion" of the process of forming a crossed product. In some cases, we obtain what we call \good" inversions, while in others we do not. For the outer-conjugacy categories, we generalize a theorem of Peder sen to obtain a fixed-point functor that is quasi-inverse to the reduced crossed-product functor for actions, and we show that this gives a good inversion. For coactions, we prove a partial version of Pedersen's the orem that allows us to define a fixed-point functor, but the question of whether it is a quasi-inverse for the crossed-product functor remains open.
AB - In this partly expository paper we compare three different categories of C*-algebras in which crossed-product duality can be for mulated, both for actions and for coactions of locally compact groups. In these categories, the isomorphisms correspond to C*-algebra isomor phisms, imprimitivity bimodules, and outer conjugacies, respectively. In each case, a variation of the fixed-point functor that arises from classical Landstad duality is used to obtain a quasi-inverse for a crossed product functor. To compare the various cases, we describe in a formal way our view of the fixed-point functor as an “inversion" of the process of forming a crossed product. In some cases, we obtain what we call \good" inversions, while in others we do not. For the outer-conjugacy categories, we generalize a theorem of Peder sen to obtain a fixed-point functor that is quasi-inverse to the reduced crossed-product functor for actions, and we show that this gives a good inversion. For coactions, we prove a partial version of Pedersen's the orem that allows us to define a fixed-point functor, but the question of whether it is a quasi-inverse for the crossed-product functor remains open.
KW - Action
KW - Category equivalence
KW - Coaction
KW - Crossed-product duality
KW - Exterior equivalence
KW - Outer conjugacy
UR - http://www.scopus.com/inward/record.url?scp=84961837248&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84961837248&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:84961837248
SN - 1076-9803
VL - 22
SP - 293
EP - 339
JO - New York Journal of Mathematics
JF - New York Journal of Mathematics
ER -