The weak heat kernel asymptotic expansion and the quantum double suspension

In this paper we are concerned with the construction of a general principle that will allow us to produce regular spectral triples with finite and simple dimension spectrum. We introduce the notion of weak heat kernel asymptotic expansion (WHKAE) property of a spectral triple and show that the weak heat kernel asymptotic expansion allows one to conclude that the spectral triple is regular with finite simple dimension spectrum. The usual heat kernel expansion implies this property. Finally we show that WHKAE is stable under quantum double suspension, a notion introduced by Hong and Szymanski. Therefore quantum double suspending compact Riemannian spin manifolds iteratively we get many examples of regular spectral triples with finite simple dimension spectrum. This covers all the odd dimensional quantum spheres. Our methods also apply to the case of noncommutative torus.