process

154 C. VoigtUsing the antipode S one obtains that every bornological quantum group is equippedwith a faithful right invariant functional as well. Again, such a functional is uniquelydetermined up to a scalar. There are injective bounded linear maps F l ; F r ; G l ; G r W H !H 0 D Hom.H; C/ defined by the formulasF l .x/.h/ D .hx/;G l .x/.h/ D .hx/;F r .x/.h/ D .xh/;G r .x/.h/ D .xh/:The images of these maps coincide and determine a vector space yH . Moreover, thereexists a unique bornology on yH such that these maps are bornological isomorphisms.The bornological vector space yH is equipped with a multiplication which is inducedfrom the comultiplication of H . In this way yH becomes an essential bornologicalalgebra and the multiplication of H determines a comultiplication on yH .Theorem 2.3. Let H be a bornological quantum group. Then yH with the structuremaps described above is again a bornological quantum group. The dual quantum groupof yH is canonically isomorphic to H .Explicitly, the duality isomorphism P W H ! yH is given by P D yG l F l S orequivalently P D yF r G r S. Here we write yG l and yF r for the maps defined aboveassociated to the dual Haar functionals on yH . The second statement of the previoustheorem should be viewed as an analogue of the Pontrjagin duality theorem.In [27] all calculations were written down explicitly in terms of the Galois mapsand their inverses. However, in this way many arguments tend to become lengthy andnot particularly transparent. To avoid this we shall use the Sweedler notation in thesequel. That is, we write.x/ D x .1/ ˝ x .2/for the coproduct of an element x, and accordingly for higher coproducts. Of coursethis has to be handled with care since expressions like the previous one only have aformal meaning. Firstly, the element .x/ is a multiplier and not contained in an actualtensor product. Secondly, we work with completed tensor products which means thateven a generic element in H y˝ H cannot be written as a finite sum of elementarytensors as in the algebraic case.3 Actions, coactions and crossed productsIn this section we review the definition of essential comodules over a bornologicalquantum group and their relation to essential modules over the dual. Moreover weconsider actions on algebras and their associated crossed products and prove an analogueof the Takesaki–Takai duality theorem.Let H be a bornological quantum group. Recall from Section 2 that a module Vover H is called essential if the module action induces an isomorphism H y˝H V Š V .A bounded linear map f W V ! W between essential H -modules is called H -linear or