Dualities for maximal coactions

TL;DR

This paper introduces a new duality construction for maximal coactions in $C^*$-algebras, utilizing Fischer's maximalizations, and formulates it within a category-theoretic framework for better manageability.

Contribution

It develops a novel crossed-product duality for maximal coactions using a generalized fixed-point algebra, and reformulates it categorically for nondegenerate *-homomorphisms and $C^*$-correspondences.

Findings

Constructed a generalized fixed-point algebra for maximal coactions.

Recovered the coaction via a double crossed product.

Outlined partial results for the outer category.

Abstract

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 $M(A\rtimes_{\delta} G \rtimes_{\widehat{\delta}} G)$, 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 $C^*$-correspondences. Also, we outline partial results for the "outer" category, studied previously by the authors.