Dualities for maximal coactions

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 $(A,\delta )$ we define a generalized fixed-point algebra as a certain subalgebra of $M(A{⋊}_{\delta }G{⋊}_{\widehat{\delta }}G)$, 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, studied previously by the authors.