Let P ⊂ A be an inclusion of σ-unital C*-algebras with a finite index in the sense of Izumi. Then we introduce the Rokhlin property for a conditional expectation E from A onto P and show that if A is simple and satisfies any of the property (1) ∼ (12) listed in the below, and E has the Rokhlin property, then so does P .Date: 9 November 2018. 2000 Mathematics Subject Classification. Primary 46L55; Secandary 46L35.

We shall introduce the notions of the strong Morita equivalence for unital inclusions of unital C * -algebras and conditional expectations from an equivalence bimodule onto its closed subspace with respect to conditional expectations from unital C * -algebras onto their unital C * -subalgebras. Also, we shall study their basic properties.Proposition 2.2. The strong Morita equivalence for inclusions of C * -algebras is equivalence relation.Proof. It suffices to show the transitivity since the other conditions clearly hold. Let A ⊂ C and B ⊂ D and K ⊂ L be inclusions of C * -algebras. We suppose that A ⊂ C is strongly Morita equivalent to B ⊂ D with respect to a C − D-equivalence bimodule Y and its closed subspace X and that B ⊂ D is strongly Morita equivalent to K ⊂ L with respect to a D − L-equivalence bimodule W and its closed subspace Z. We consider the closed subspace of Y ⊗ D W spanned by the setWe denote it by X ⊗ D Z. For any x 1 , x

We shall introduce the approximate representability and the Rohlin property for coactions of a finite dimensional C * -Hopf algebra on a unital C * -algebra and discuss some basic properties of approximately representable coactions and coactions with the Rohlin property of a finite dimensional C * -Hopf algebra on a unital C * -algebra. Also, we shall give an example of an approximately representable coaction of a finite dimensional C * -Hopf algebra on a simple unital C * -algebra which has also the Rohlin property and we shall give the 1-cohomology vanishing theorem for coactions of a finite dimensional C * -Hopf algebra on a unital C * -algebra and the 2-cohomology vanishing theorem for twisted coactions of a finite dimensional C * -Hopf algebra on a unital C * -algebra. Furthermore, we shall introduce the notion of the approximately unitary equivalence of coactions of a finite dimensional C * -Hopf algebra H on a unital C * -algebra A and show that if ρ and σ, coactions of H on a separable unital C * -algebra A, which have the Rohlin property, are approximately unitarily equivalent, then there is an approximately inner automorphism α on A such thatLet ρ be a coaction of H on A and A ρ the fixed point C * -subalgebra of A for ρ, that is,Let E ρ be the canonical conditional expectation from A onto A ρ defined by E ρ (a) = τ · ρ a = (id ⊗ τ )(ρ(a)) for any a ∈ A. We note that E ρ is faithful by [10, Proposition 2.12].Definition 2.3. We say that ρ is saturated if the action of H 0 on A induced by ρ is saturated in the sense of [10].In Sections 4, 5 and 6 of [6], we suppose that the action of H on A is saturated. But, without saturation, all the statements in Sections 4 and 5 and Theorem 6.4 of [6] hold. Hence we obtain the following proposition.Proof. Since the dual coaction of a twisted coaction is saturated, this is immediate by Proposition 2.3. DualityLemma 3.1. With the above notations, V I V * J = 1 ⋊ ρ τ if I = J 0 if I = J. Proof. Let I = (i, j, k) and J = (s, t, r) be any elements in Λ. Then 1 0 )V L = Ψ([a IJ ][b IJ ]) by Lemma 3.1. For any [a IJ ] ∈ M N (A), Ψ([a IJ ])

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Following Jansen and Waldmann, and Kajiwara and Watatani, we shall introduce notions of coactions of a finite dimensional C * -Hopf algebra on a Hilbert C * -bimodule of finite type in the sense of Kajiwara and Watatani and define their crossed product. We shall investigate their basic properties and show that the strong Morita equivalence for coactions preserves the Rohlin property for coactions of a finite dimensional C * -Hopf algebra on unital C * -algebras.2010 Mathematics Subject Classification. Primary 46L05; Secondary 46L08.