Equivariant Picard groups of C*-algebras with finite dimensional C*-Hopf algebra coactions

الأصل · EN

Let A be a C*-algebra and H a finite dimensional C*-Hopf algebra with its dual C*-Hopf algebra H⁰. Let (ρ, u) be a twisted coaction of H⁰ on A. We shall define the (ρ, u, H)-equivariant Picard group of A, which is denoted by ₕρ, ᵘ(A), and discuss basic properties of ₕρ, ᵘ(A). Also, we suppose that (ρ, u) is the coaction of H⁰ on the unital C*-algebra A, that is, u=1⊗ 1⁰. We investigate the relation between (Aˢ), the ordinary Picard group of Aˢ and ₕρˢ(Aˢ) where Aˢ is the stable C*-algebra of A and ρˢ is the coaction of H⁰ on Aˢ induced by ρ. Furthermore, we shall show that ₕ₀ρ(Aᵨ, ᵤH) is isomorphic to ₕρ, ᵘ(A), where ρ is the dual coaction of H on the twisted crossed product Aᵨ, ᵤH of A by the twisted coaction (ρ, u) of H⁰ on A.