a b s t r a c tIn [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], we presented a K -theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SU q (2) and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of the modular group of the Haar state. In contrast to the Cuntz algebras studied in [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], the computations are considerably more complex and interesting, because there are non-trivial 'eta' contributions to this index.

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The construction of public morals occupies an essential position in the construction of modern communities. The transformation of transformed communities means the reconstruction of public morals. In the transition from a transformed community to a modern community, this article starts from the historical research context of public moral theory and combines community participation and governance theory from the perspective of public morality. Different from the passive participation image and macro-narrative Research of community residents or villagers in many studies, this article focuses on micro-level case analysis; in specific geographical regions and administrative regions of historical evolution, fragmented narrative methods are adopted by community residents' Interviews and cases to explore the changes in publicity from a micro perspective, pointing out that in a general community where some of the geographical bonds are partially preserved and public morals are generally declining, the emotional bonds are relinked utilizing geographical consensus reconstruction, public cultural cultivation, and welfare mobilization, Public interest and private interest reconcile and ultimately stimulate the vitality of community participation. Therefore, this research will contribute to the application of public moral theory in practice to a certain extent, and it will help to study the publicity of the community at the micro level and explore the way to govern publicity at the level of public morals.