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Coactions of a finite-dimensional \(C^*\)-Hopf algebra on unital \(C^*\)-algebras, unital inclusions of unital \(C^*\)-algebras and strong Morita equivalence. (English) Zbl 1466.46045

Summary: 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 \((\rho,u)\) and \((\sigma,v)\) be twisted coactions of \(H^0\) on \(A\) and \(B\), respectively. In this paper, we show the following theorem: Suppose that the unital inclusions \(A\subset A\rtimes_{\rho,u}H\) and \(B\subset B\rtimes_{\sigma,v}H\) are strongly Morita equivalent. If \(A'\cap(A\rtimes_{\rho,u}H)=\mathbb{C}1\), then there is a \(C^*\)-Hopf algebra automorphism \(\lambda^0\) of \(H^0\) such that the twisted coaction \((\rho,u)\) is strongly Morita equivalent to the twisted coaction \(((\mathrm{id}_B\otimes\lambda^0)\circ\sigma$, $(\mathrm{id}_B\otimes\lambda^0 \otimes\lambda^0)(v))\) induced by \((\sigma,v)\) and \(\lambda^0\).

MSC:

46L05 General theory of \(C^*\)-algebras
46L08 \(C^*\)-modules
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References:

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