process

156 C. VoigtProof. Let V be the space V equipped with the trivial H -action induced by the counit.We have a natural H -linear isomorphism ˛l W H y˝ V ! H y˝ V given by ˛l.x ˝v/ D x .1/ ˝ S.x .2/ / v. Similarly, the map ˛r W V y˝ H ! V y˝ H given by˛r.v ˝ x/ D S 1 .x .1/ / v ˝ x .2/ is an H -linear isomorphism. Since H is projectivethis yields the claim.Using category language an H -algebra is by definition an algebra in the categoryH -Mod. We formulate this more explicitly in the following definition.Definition 3.4. Let H be a bornological quantum group. An H -algebra is a bornologicalalgebra A which is at the same time an essential H -module such that the multiplicationmap A y˝ A ! A is H -linear.If A is an H -algebra we will also speak of an action of H on A. Remark thatwe do not assume that an algebra has an identity element. The unitarization A C ofan H -algebra A becomes an H -algebra by considering the trivial action on the extracopy C.According to Theorem 3.2 we can equivalently describe an H -algebra as a bornologicalalgebra A which is at the same time an essential yH -comodule such that the multiplicationis yH -colinear.Under additional assumptions there is another possibility to describe this structurewhich resembles the definition of a coaction in the setting of C -algebras. Let us callan essential bornological algebra A regular if it is equipped with a faithful boundedlinear functional and satisfies the approximation property. If A is regular it followsfrom [27] that the natural bounded linear map A y˝ H ! M.A y˝ H/is injective.Definition 3.5. Let H be a bornological quantum group. An algebra coaction of Hon a regular bornological algebra A is an essential algebra homomorphism ˛ W A !M.A y˝ H/such that the coassociativity condition.˛ y˝ id/˛ D .id y˝ /˛holds and the maps ˛l and ˛r from A y˝ H to M.A y˝ H/given by˛l.a ˝ x/ D .1 ˝ x/˛.a/; ˛r.a ˝ x/ D ˛.a/.1 ˝ x/induce bornological automorphisms of A y˝ H .It can be shown that an algebra coaction ˛ W A ! M.A y˝ H/on a regular bornologicalalgebra A satisfies .id y˝ /˛ D id. In particular, the map ˛ is always injective.Proposition 3.6. Let H be a bornological quantum group and let A be a regularbornological algebra. Then every algebra coaction of yH on A corresponds to a uniqueH -algebra structure on A and vice versa.Proof. Assume that ˛ is an algebra coaction of yH on A and define D ˛r. Bydefinition is a right yH -linear automorphism of A y˝ yH . Moreover we have for a 2 A