Abstract
The notion of H-covariant strong Morita equivalence is introduced for * -algebras over C = R ( i ) with an ordered ring R which are equipped with a * -action of a Hopf * -algebra H. This defines a corresponding H-covariant strong Picard groupoid which encodes the entire Morita theory. Dropping the positivity conditions one obtains H-covariant * -Morita equivalence with its H-covariant * -Picard groupoid. We discuss various groupoid morphisms between the corresponding notions of the Picard groupoids. Moreover, we realize several Morita invariants in this context as arising from actions of the H-covariant strong Picard groupoid. Crossed products and their Morita theory are investigated using a groupoid morphism from the H-covariant strong Picard groupoid into the strong Picard groupoid of the crossed products.
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