The \(\mathrm{C}^\ast\)-algebra index for observable algebra in non-equilibrium Hopf spin models (Q2086127)

Let \(K\) be a Hopf \(*\)-subalgebra of a finite dimensional Hopf C\(^*\)-algebra \(H\). Given the observable algebra \(\mathcal{A}_K\) in non-equilibrium spin models, in which \(\mathcal{A}_K\) is endowed with a coaction of relative quantum double \(D(H,K)\), one can construct the field algebra \(\mathcal{F}_K=\mathcal{A}_K\rtimes \widehat{D(H,K)}\), here \(\widehat{D(H,K)}\) is the dual of \(D(H,K)\). In this article, the authors shows that the Haar integral of \(D(H,K)\) yields a a faithful conditional expectation from \(\Gamma: \mathcal{F}_K\to \mathcal{A}_K\) and compute its index.