Conjugation in Representation Categories of Multiplicative Unitaries and Their Actions on C*-Algebras

Abstract

A conjugation functor F on a full subcategory of R(V ), the representation category of a multiplicative unitary V, is defined. If V has a conjugate, it is also regular and the domain of F is all R(V ). Examples of selfconjugate multiplicative unitaries are discussed. A coaction of the Hopf C*-algebra associated with V on the Cuntz algebra O d is canonically defined by a unitary object W of R(V ) acting on a d-dimensional Hilbert space. As in the group action case if d= and W belongs to the domain of F, ergodic coactions are often characterized by the absence of finite dimensional subrepresentations of W. Furthermore model actions of compact quantum group duals on C*-algebras are defined.

An Algebraic Duality Theory for Multiplicative Unitaries

“…. Now the arguments of [14]; Proposition 6.5, except that ρ K (H s ) * λ ϑH,K W (H r , H s )ρ K (H r ) ⊆ M (A(V )) here, show that E is the desired conditional expectation.…”

“…Here in the definition of λ ϑH,K W we consider H and K as Hilbert spaces of support I in some B(H) and regard ϑ H,K W as mapping H onto ϑ H,K W H. The arguments of [5] or [14]; Section 6 generalize to show that the monomorphism…”

Regular Objects, Multiplicative Unitaries and Conjugation

“…We shall see later that the left regular representation of a locally compact quantum group is selfconjugate in this sense. Other examples arise from compact quantum groups [15], Hopf-von Neumann algebras [4] and Kac systems [1] as pointed out in [11]. It raises the question of whether the existence of conjugate representations might prove a more effective postulate than regularity in developing the theory of multiplicative unitaries.…”