A Note on Spectral Triples and Quasidiagonality

The concept of a spectral triple (unbounded Fredholm module) due to A.Connes ([Co1]) is a natural noncommutative generalisation of a notion of a compact manifold, with certain summability properties corresponding in the classical case to the dimension of the manifold. Recently E.Christensen and C. Ivan established the existence of spectral triples on arbitrary AF algebras ([CI]). In this note we generalise their result to arbitrary quasidiagonal (representations of) C-algebras. Contrary to the AF situation our triples might in general have bad summability properties and it is not clear whether they satisfy Rieffel’s condition (i.e. whether the topology they induce on the state space coincides with the weak-topology). Although the connections between properties related to quasi-diagonality and the existence of unbounded Fredholm modules seem to have been known for a long time (see for example [Vo1]), explicit constructions have been until now given only in presence of a filtration of the C-algebra in question consisting of finite-dimensional subspaces ([Vo1], [CI]). We also show that the existence of spectral triples of compact type does not imply quasidiagonality by exhibiting a simple example of such a triple (with bad summability properties) on the natural, non-quasidiagonal, representation of the Toeplitz algebra.