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In [CPR2], we presented a K-theoretic approach to finding invariants ofalgebras with no non-trivial traces. This paper presents a new example that ismore typical of the generic situation. This is the case of an algebra thatadmits only non-faithful traces, namely SU_q(2), and also KMS states. Our mainresults are index theorems (which calculate spectral flow), one using ordinarycyclic cohomology and the other using twisted cyclic cohomology, where thetwisting comes from the generator of the modular group of the Haar state. Incontrast to the Cuntz algebras studied in [CPR2], the computations areconsiderably more complex and interesting, because there are nontrivial `eta'contributions to this index.

Spectral flow invariants and twisted cyclic theory from the Haar state on SU_q(2)