In this paper we consider algebras with involution over a ring C which is given by the quadratic extension by i of an ordered ring R. We discuss the -representation theory of such -algebras on pre-Hilbert spaces over C and develop the notions of Rieffel induction and formal Morita equivalence for this category analogously to the situation for C-algebras. Throughout this paper the notion of posi...

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 groupo...

We develop Morita theory for finitary additive 2-representations of finitary 2-categories. As an application we describe Morita equivalence classes for 2-categories of projective functors associated to finite dimensional algebras and for 2-categories of Soergel bimodules.