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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. [ABSTRACT FROM AUTHOR]

We prove that if E and F are large ideals of B(G) for which the associated coaction functors are exact, then the same is true for E ∩ F. We also give an example of a coaction functor whose restriction to the maximal coactions does not come from any large ideal. [ABSTRACT FROM AUTHOR]

The first imprimitivity theorems identified the representations of groups or dynamical systems which are induced from representations of a subgroup. Symmetric imprimitivity theorems identify pairs of crossed products by different groups which are Morita equivalent and hence have the same representation theory. Here we consider commuting actions of groups H and K on a C*-algebra which are saturated and proper as defined by Rieffel in 1990. Our main result says that the resulting Morita equivalence of crossed products is natural in the sense that it is compatible with homomorphisms and induction processes. [ABSTRACT FROM AUTHOR]

Cuntz and Li have defined a C*-algebra associated to any integral domain, using generators and relations, and proved that it is simple and purely infinite and that it is stably isomorphic to a crossed product of a commutative C*-algebra. We give an approach to a class of C*-algebras containing those studied by Cuntz and Li, using the general theory of C*-dynamical systems associated to certain semidirect product groups. Even for the special case of the Cuntz–Li algebras, our development is new. [ABSTRACT FROM AUTHOR]