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DUALITIES FOR MAXIMAL COACTIONS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 102 / Issue 2 / April 2017
- Published online by Cambridge University Press:
- 12 May 2016, pp. 224-254
- Print publication:
- April 2017
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- Article
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We present a new construction of crossed-product duality for maximal coactions that uses Fischer’s work on maximalizations. Given a group
$G$ and a coaction 
$(A,\unicode[STIX]{x1D6FF})$ we define a generalized fixed-point algebra as a certain subalgebra of 
$M(A\rtimes _{\unicode[STIX]{x1D6FF}}G\rtimes _{\,\widehat{\unicode[STIX]{x1D6FF}}}G)$, and recover the coaction via this double crossed product. Our goal is to formulate this duality in a category-theoretic context, and one advantage of our construction is that it breaks down into parts that are easy to handle in this regard. We first explain this for the category of nondegenerate *-homomorphisms and then, analogously, for the category of 
$C^{\ast }$-correspondences. Also, we outline partial results for the ‘outer’ category, which has been studied previously by the authors.
