0 Examples of amenable Kac system

By giving an interesting characterisation of amenable multiplicative unitaries, we show in a simple way that bicrossproducts of amenable locally compact groups is both amenable and coamenable. Amenability of Kac algebras has been studied in [3]. In [1], amenability of regular multiplicative unitaries was also defined. Recently, we studied in [7] and [9] amenable Hopf C *-algebras and amenable Kac algebras. However, there wasn't any non-trivial examples of amenable Kac algebras explicitly stated in any literature so far. In this note, we will give a simple proof for the following expected statement (we do not know if another proof for this statement is already known – note that the " reduced algebra " of the bicrossproduct is the reduced crossed product, one would imagine the " full algebra " to be the full crossed product but it seems not clear to us whether this is true): in the construction of the " bicrossproduct " (in [2, §1]) of two locally compact groups, if one of the groups is amenable, then the " bicrossproduct " is amenable (or coamenable – depended on the convention). In fact, we prove this by using a simple but seeming powerful characterisation for amenable multiplicative unitaries (Proposition 3). Let us first recall from [6, 2.1 & 2.2(b)] as well as [1, A.13(c)] the following definitions. Definition 1 (a) Let V be a multiplicative unitary on H (in the sense of [1, 1.1]). Then V is called a C *-multiplicative unitary if for any representation X and co-representation Y of V (see [1, A.1]), the setsˆS X = {(id ⊗ ω)(X) : ω ∈ L(H) * } and S Y = {(ω ⊗ id)(Y) : ω ∈ L(H) * } are C *-algebras such that X ∈ M (ˆ S X ⊗ S V) and Y ∈ M (ˆ S V ⊗ S Y). (Recall that in this case, S V andˆS V are Hopf C *-algebras with coproducts δ andˆδ defined by the formulas in [1, 3.8].) (b) Let V be a C *-multiplicative unitary. For any C *-algebra A, a unitary U ∈ M (A ⊗ S V) is called a unitary corepresentation if (id ⊗ δ)(U) = U 12 U 13 (recall that unitary corepresentations of S V corresponds directly to representations of V). Moreover, there exists an universal object (ˆ S p , V ′) for unitary corepresentations of S in the sense …