Inclusions of unital $C^*$-algebras of index-finite type with depth 2 induced by saturated actions of finite dimensional $C^*$-Hopf algebras

Abstract

Let B be a unital C * -algebra and H a finite dimensional C * -Hopf algebra with its dual C * -Hopf algebra H 0 . We suppose that there is a saturated action of H on B and we denote by A its fixed point C * -subalgebra of B. Let E be the canonical conditional expectation from B onto A. In the present paper, we shall give a necessary and sufficient condition that there are a weak action of H 0 on A and a unitary cocycle σ of H 0 ⊗ H 0 to A satisfying that there is an isomorphism π of A σ H 0 onto B, which is the twisted crossed product of A by the weak action of H 0 on A and the unitary cocycle σ , such that F = E • π, where F is the canonical conditional expectation from A σ H 0 onto A.

The Rohlin property for coactions of finite dimensional $C^*$-Hopf algebras on unital $C^*$-algebras

“…By Sweedler [9, pp. 69-70] it becomes a unital * -algebra which is also defined in [6,Sections 2 and 3]. In the same way as [6, Sections 2 and 3], we define a unital * -algebra Hom(H 0 ⊗ H 0 , A).…”

“…We call the pair of the weak action and the unitary cocycle u the twisted action of H 0 on A induced by (ρ, u). By [6,Section 3], we can construct the twisted crossed product of A by H 0 which is denoted by A ⋊ ρ,u H 0 . Let ρ be the dual coaction of ρ, which is defined by for any a ∈ A, φ ∈ H 0 , ρ(a ⋊ ρ,u φ) = (a ⋊ ρ,u φ (1) ) ⊗ φ (2) , where a ⋊ ρ,u φ denotes the element in A ⋊ ρ,u H 0 induced by a ∈ A and φ ∈ H 0 .…”

“…Let V be an element in (A ⋊ ρ,u H 0 ) ⊗ H induced by V . By [6,Lemma 3.12], we can see that V and V are unitary elements in (A ⋊ ρ,u H 0 ) ⊗ H and Hom(H 0 , A ⋊ ρ,u H 0 ), respectively and that…”

“…The restriction of θ to 1 ⋊ H, the C * -subalgebra of Proof. By Lemma 9.2 and [10, Theorem 2.2], for any h ∈ H, 3) , S(h (4) ) * ) * ⋊ ρ,u S(h (2) )) = i,j,k,j1,j2,j3,j4,i1 (5) ), h (6) )] u(h (3) , S(h (4) )).…”

The strong Morita equivalence for coactions of a finite-dimensional C*-Hopf algebra on unital C*-algebras

“…Let V ρ be a linear map from H to A ⋊ ρ,u H defined by V ρ (h) = 1 ⋊ ρ,u h for any h ∈ H. By [10], V ρ is a unitary element in Hom(H, A⋊ ρ,u H). Let V ρ be the unitary element in (A ⋊ ρ,u H) ⊗ H 0 induced by V ρ .…”