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 (At,K,Dα,β) 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 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 (α, 1) → (0, 1), while the basic spectral triple (A, H,D) is obtained from this family under a sort of a dual limiting process for (1, β) → (1, 0). We show that our constructions will provide families of spectral triples for the unitarized compacts and for the Podles̀ sphere. In the case of the compacts we investigate to which extent our proposed spectral triple satisfies Connes’ 7 axioms for noncommutative geometry, [8]. Introduction The so called Toeplitz algebra, say T , may be obtained in a number of different ways. The most simple description of it is possibly as the C*-algebra on the Hilbert space l(N) generated by the unilateral shift. A more profound description which relates to analysis, can be obtained via the algebra, C := C(T), of continuous functions on the unit circle. A function f in this algebra is represented as a multiplication operator, Mf on the Hilbert space H := L (T) of square integrable functions. This space has a subspace H+, which consists of those functions in H that have an analytic extension to the interior of the unit disk. Let P+ Date: July 19, 2008. 1991 Mathematics Subject Classification. Primary, 58B34, 46L65 ; Secondary, 46L87 , 83C65 .