process

158 C. VoigtTo every H -algebra A one may form the associated crossed product A Ì H . Theunderlying bornological vector space of A Ì H is A y˝ H and the multiplication isdefined by the chain of mapsA y˝ H y˝ A y˝ H 24r A y˝ H y˝ A y˝ H id y˝ y˝id A y˝ A y˝ H y˝id A y˝ Hwhere denotes the action of H on A. Explicitly, the multiplication in A Ì H is givenby the formula.a Ì x/.b Ì y/ D ax .1/ b ˝ x .2/ yfor a; b 2 A and x;y 2 H . On the crossed product A Ì H one has the dual action ofyH defined byf .a Ì x/ D a Ì .f * x/for all f 2 yH . In this way AÌH becomes an yH -algebra. Consequently one may formthe double crossed product AÌH Ì yH . In the remaining part of this section we discussthe Takesaki–Takai duality isomorphism which clarifies the structure of this algebra.First we describe a general construction which will also be needed later in connectionwith stability of equivariant cyclic homology. Assume that V is an essential H -moduleand that A is an H -algebra. Moreover let b W V V ! C be an equivariant boundedlinear map. We define an H -algebra l.bI A/ by equipping the space V y˝ A y˝ V withthe multiplication.v 1 ˝ a 1 ˝ w 1 /.v 2 ˝ a 2 ˝ w 2 / D b.w 1 ;v 2 /v 1 ˝ a 1 a 2 ˝ w 2and the diagonal H -action.As a particular case of this construction consider the space V D yH with the regularaction of H given by .t * f /.x/ D f.xt/and the pairingˇ.f; g/ D O .fg/:We write K H for the algebra l.ˇI C/ and A y˝ K H for l.ˇI A/. Remark that the actionon A y˝ K H is not the diagonal action in general. We denote an element f ˝ a ˝ g inthis algebra by jf i˝a ˝hgj in the sequel. Using the isomorphism yF r S 1 W yH ! Hwe identify the above pairing with a pairing H H ! C. The corresponding actionof H on itself is given by left multiplication and using the normalization D S. /we obtain the formulaˇ.x; y/ D ˇ.SG r S.x/;SG r S.y// D ˇ.F l .x/; F l .y// D .S 1 .y/x/ D .S.x/y/for the above pairing expressed in terms of H .Let H be a bornological quantum group and let A be an H -algebra. We define abounded linear map A W A Ì H Ì yH ! A y˝ K H by A .a Ì x Ì F l .y// Djy .1/ S.x .2/ /i˝y .2/ S.x .1/ / a ˝hy .3/ jand it is easily verified that A is an equivariant bornological isomorphism. In addition, astraightforward computation shows that A is an algebra homomorphism. Consequentlywe obtain the following analogue of the Takesaki–Takai duality theorem.