Saturated Actions of Finite Dimensional Hopf *-Algebras on $C^*$-Algebras.

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

“…Following [14], we shall state the definition of a finite dimensional C * -Hopf algebra and its basic properties. Throughout this paper, H denotes a finite dimensional C * -Hopf algebra.…”

“…Also, we can define a conditional expectation E from A onto A H with Index(E) = dim(H ) by E(x) = e · x for x ∈ A by [14], where Index(E) is the Watatani index of E. We call E the canonical conditional expectation of A onto A H .…”

“…Let A be the fixed point C * -subalgebra B H and E the canonical conditional expectation from B onto A. Let B H be the crossed product of B by the action of H on B, which is defined in [14]. In [14], they showed that B H is isomorphic to B 1 , the C * -basic construction induced by E. Let ρ be the coaction of B 1 to B 1 ⊗H defined by ρ(b h) = (h) (b h (1) ) ⊗ h (2) for b ∈ B and h ∈ H , where we identify B 1 with B H and (h) = (h) h (1) ⊗ h (2) , is the comultiplication of H .…”

“…Let B H be the crossed product of B by the action of H on B, which is defined in [14]. In [14], they showed that B H is isomorphic to B 1 , the C * -basic construction induced by E. Let ρ be the coaction of B 1 to B 1 ⊗H defined by ρ(b h) = (h) (b h (1) ) ⊗ h (2) for b ∈ B and h ∈ H , where we identify B 1 with B H and (h) = (h) h (1) ⊗ h (2) , is the comultiplication of H . Our main result, Theorem 6.4, is that B can be represented by the twisted crossed product A σ H 0 if and only if ρ(e A ) and e A ⊗ 1 are Murray-von Neumann equivalent, written ρ(e A ) ∼ e A ⊗ 1 in B 1 ⊗ H , where H 0 is the dual C * -Hopf algebra of H and e A is the Jones projection for A ⊂ B.…”

“…Modifying [14] and [1], we shall define a twisted crossed product of a unital C * -algebra by a finite dimensional C * -Hopf algebra. Let H be a finite dimensional C * -Hopf algebra and A a unital C * -algebra.…”