 

S. Kaliszewski, Tron Omland, and John Quigg
Three versions of categorical crossedproduct duality view print


Published: 
March 17, 2016

Keywords: 
action, coaction, crossedproduct duality, category equivalence, C*correspondence, exterior equivalence, outer conjugacy 
Subject: 
Primary 46L55; Secondary 46M15 


Abstract
In this partly expository paper we compare three different categories of C*algebras
in which crossedproduct duality can be formulated,
both for actions and for coactions of locally compact groups.
In these categories, the isomorphisms correspond to
C*algebra isomorphisms, imprimitivity bimodules,
and outer conjugacies, respectively.
In each case, a variation of the fixedpoint functor that arises from classical
Landstad duality is used to obtain a quasiinverse for a crossedproduct functor.
To compare the various cases,
we describe in a formal way
our view of the fixedpoint 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 outerconjugacy categories,
we generalize a theorem of Pedersen to obtain a fixedpoint functor
that is quasiinverse to the reducedcrossedproduct functor for actions,
and we show that this gives a good inversion.
For coactions, we prove a partial version of Pedersen's theorem
that allows us to define a fixedpoint functor, but
the question of whether it is a
quasiinverse for the crossedproduct functor remains open.


Acknowledgements
The second author is funded by the Research Council of Norway (Project no.: 240913).


Author information
S. Kaliszewski:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287
kaliszewski@asu.edu
Tron Omland:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287
omland@asu.edu
John Quigg:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287
quigg@asu.edu

