[Paper Review] Coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras, unital inclusions of unital $C^*$-algebras and the strong Morita equivalence

Kazunori Kodaka, Tamotsu Teruya|arXiv (Cornell University)|Jun 29, 2017

Advanced Operator Algebra Research11 references3 citations

TL;DR

This paper establishes a converse to a known result in operator algebras: if the unital inclusions $ A \subset A \rtimes_{\rho,u}H $ and $ B \subset B\rtimes_{\sigma,v}H $ induced by twisted coactions of a finite-dimensional $ C^* $-Hopf algebra $ H^0 $ on unital $ C^* $-algebras $ A $ and $ B $ are strongly Morita equivalent, and if the relative commutant $ A' \cap (A\rtimes_{\rho,u}H) = \mathbb{C}1 $, then the twisted coactions $ (\rho,u) $ and $ (\sigma,v) $ are strongly Morita equivalent up to an automorphism of $ H^0 $. The result provides a classification of twisted coactions via strong Morita equivalence of crossed products.

ABSTRACT

Let $A$ and $B$ be unital $C^*$-algebras and let $H$ be a finite dimensional $C^*$-Hopf algebra. Let $H^0$ be its dual $C^*$-Hopf algebra. Let $(ρ, u)$ and $(σ, v)$ be twisted coactions of $H^0$ on $A$ and $B$, respectively. In this paper, we shall show the following theorem: We suppose that the unital inclusions $A\subset A times_{ρ, u}H$ and $B\subset B times_{σ, v}H$ are strongly Morita equivalent. If $A'\cap (A times_{ρ, u}H)=\BC1$, then there is a $C^*$-Hopf algebra automorphism $λ^0$ of $H^0$ such that the twisted coaction $(ρ, u)$ is strongly Morita equivalent to the twisted coaction $((\id_B \otimesλ^0 )\circσ\, , \, (\id_B \otimesλ^0 \otimesλ^0 )(v))$ induced by $(σ, v)$ and $λ^0$.

Motivation & Objective

• To investigate the inverse relationship between strong Morita equivalence of unital inclusions $ A \subset A\rtimes_{\rho,u}H $ and $ B \subset B\rtimes_{\sigma,v}H $, and strong Morita equivalence of the underlying twisted coactions $ (\rho,u) $ and $ (\sigma,v) $.
• To determine when strong Morita equivalence of crossed product inclusions implies equivalence of the corresponding twisted coactions, up to automorphism of the dual $ C^* $-Hopf algebra.
• To establish a classification result for twisted coactions on unital $ C^* $-algebras via the strong Morita equivalence class of their crossed products.
• To extend previous results on strong Morita equivalence of twisted coactions by proving a converse implication under a non-degeneracy condition on the relative commutant.

Proposed method

• The authors use the duality between $ H $ and $ H^0 $, identifying twisted coactions with dual actions via the Fourier transform on $ H $.
• They apply the theory of Hilbert $ C^* $-bimodules and strong Morita equivalence of inclusions, relying on [11, Definition 2.1] and the existence of equivalence bimodules.
• The key technique involves constructing an isomorphism $ \pi $ between crossed products $ A\rtimes_{\rho,u}H $ and $ A\rtimes_{\rho_{\lambda^0},u_{\lambda^0}}H $ induced by a $ C^* $-Hopf algebra automorphism $ \lambda^0 $ of $ H^0 $, showing that such automorphisms preserve the Morita class of the inclusion.
• They use the dual coaction $ \widehat{\rho} $ of $ (\rho,u) $ and relate it to the dual of the twisted coaction induced by $ \lambda^0 $, leveraging results from [10, Corollary 4.8].
• The proof relies on Lemmas 5.3 and 6.1 to relate the dual coaction to the twisted coaction under $ \lambda^0 $, and uses the isomorphism $ \pi $ to transfer Morita equivalence between inclusions.
• The condition $ A' \cap (A\rtimes_{\rho,u}H) = \mathbb{C}1 $ ensures that the inclusion is non-degenerate and allows the construction of a unique automorphism $ \lambda^0 $ linking the coactions.

Experimental results

Research questions

• RQ1 Under what conditions does strong Morita equivalence of the unital inclusions $ A \subset A\rtimes_{\rho,u}H $ and $ B \subset B\rtimes_{\sigma,v}H $ imply strong Morita equivalence of the twisted coactions $ (\rho,u) $ and $ (\sigma,v) $?
• RQ2 Can the inverse of the known implication (that Morita equivalence of coactions implies Morita equivalence of inclusions) be established in the context of finite-dimensional $ C^* $-Hopf algebras?
• RQ3 Is there a canonical automorphism of the dual $ C^* $-Hopf algebra $ H^0 $ that relates two twisted coactions whose crossed product inclusions are strongly Morita equivalent?
• RQ4 How does the relative commutant condition $ A' \cap (A\rtimes_{\rho,u}H) = \mathbb{C}1 $ constrain the structure of the coaction and enable the classification of coactions via Morita equivalence?
• RQ5 To what extent is the strong Morita equivalence class of a twisted coaction determined by the strong Morita equivalence class of its crossed product inclusion?

Key findings

• If the unital inclusions $ A \subset A\rtimes_{\rho,u}H $ and $ B \subset B\rtimes_{\sigma,v}H $ are strongly Morita equivalent and $ A' \cap (A\rtimes_{\rho,u}H) = \mathbb{C}1 $, then there exists a $ C^* $-Hopf algebra automorphism $ \lambda^0 $ of $ H^0 $ such that $ (\rho,u) $ is strongly Morita equivalent to the twisted coaction $ ((\mathrm{id}_B \otimes \lambda^0) \circ \sigma, (\mathrm{id}_B \otimes \lambda^0 \otimes \lambda^0)(v)) $.
• The automorphism $ \lambda^0 $ is induced by a $ C^* $-Hopf algebra automorphism $ \lambda $ of $ H $, satisfying $ \lambda^0(\psi)(h) = \psi(\lambda^{-1}(h)) $ for all $ \psi \in H^0 $, $ h \in H $.
• The result establishes a converse to the known implication: strong Morita equivalence of inclusions implies strong Morita equivalence of coactions up to automorphism of $ H^0 $, under the non-degeneracy condition.
• The proof constructs an isomorphism $ \pi $ between crossed products that restricts to the identity on $ A $, showing that the inclusion $ A \subset A\rtimes_{\rho,u}H $ is strongly Morita equivalent to its twist via $ \lambda^0 $.
• The equivalence of the inclusions and the coactions is established through the dual coaction and the use of Hilbert $ C^* $-bimodules, with the key role played by the condition $ A' \cap (A\rtimes_{\rho,u}H) = \mathbb{C}1 $ in ensuring uniqueness of the automorphism $ \lambda^0 $.
• Corollary 6.5 establishes the equivalence of two conditions: (1) strong Morita equivalence of the inclusions, and (2) existence of an automorphism $ \lambda^0 $ such that the twisted coactions are strongly Morita equivalent via the induced coaction.