Extensions and Degenerations of Spectral Triples

For a unital C*-algebra A, which is equipped with a spectral triple (A, H, D) and an extension T of A by the compacts, we construct a two parameter family of spectral triples (A(t), K, D-alpha,D-beta) associated to T. Using Rieffel's notation of quantum Gromov-Hausdorff distance between compact quantum metric spaces it is possible to define a metric on this family of spectral triples, and we show that the distance between a pair of spectral triples varies continuously with respect to the parameters. It turns out that a spectral triple associated to the unitarization of the algebra of compact operators is obtained under the limit - in this metric - for (alpha, 1) -> (0, 1), while the basic spectral triple (A, H, D) is obtained from this family under a sort of dual limiting process for (1, beta) -> (1, 0). We show that our constructions will provide families of spectral triples for the unitarized compacts and for the Podles sphere. In the case of the compacts we investigate to which extent our proposed spectral triple satisfies Connes' 7 axioms for noncommutative geometry, [8].