The Weak Heat Kernel Expansion and the Quantum Double Suspension

We are interested in producing spectral triples that satisfy the hypothesis of the local index formula namely regularity and finiteness of the dimension spectrum. We begin with showing that  the weak heat kernel expansion is stable under quantum double suspension and the usual heat kernel expansion implies weak heat kernel expansion. Therefore quantum double suspending ordinary compact spin manifolds  we get spectral triples that satisfy weak heat kernel expansion. We further show that weak heat kernel expansion implies regularity and finiteness of the dimension spectrum. Therefore we get lots of examples of spectral triples satisfying the hypothesis of the local index formula. This covers the examples coming from quantum spheres.