An Algebraic Duality Theory for Multiplicative Unitaries

12, 415 - 459, 2001 Multiplicative unitaries aredescribed in terms of apair of commuting shifts of relativedepth two. They can be generated from ambidextrous Hilbert spaces inatensor C*–category. The algebraic analogue of theTakesaki-Tatsuuma Duality Theorem characterizes abstractly C*–algebras acted on by unital endomorphisms that are intrinsically related to the regular representation of a multiplicative unitary. The relevant C∗–algebras turn out to be simple and indeed separable if the corresponding multiplicative unitaries act onaseparable Hilbert space. A categorical analogue provides internal characterizations of minimal representation categories of a multiplicative unitary. Endomorphisms of the Cuntz algebra rlated algebraically to the grading are discussed as is the notionof braidedsymmetry in a tensor C*-category.