2 7 Se p 20 07 EXTENDED SPECTRAL TRIPLES AND DEFORMATIONS

For a unital C*-algebra A, which is equipped with a spectral triple (A,H,D) and a Toeplitz extension, T , of A by the compacts, we construct a two parameter family of spectral triples (At,K,D(α,β)) associated to T . Using Rieffel’s notation, the family of spectral triples induce a two parameter family of compact quantum metric spaces, and we study the variation of the quantum Gromov-Hausdorff distances between these spaces with respect to the parameters. It turns out that both the C*-algebra A and the unitarization of the algebra of compact operators can be obtained as limits when the parameters follow simple paths in the parameter space. 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+ denote the orthogonal projection of H onto H+, then the compression, to H+ of a multiplication operatorMf for a continuous function f on T becomes the Toeplitz operator Tf := P+Mf |H+. The mapping C ∋ f → Tf relates to the differentiable structure on the circle in the way, that for the ordinary differentiation on the circle with respect to arc length, i. e. D := 1 i d dθ , we know that the space H+ is the closed linear span of the eigenvectors corresponding to non negative eigenvalues for D, so Date: July 24, 2008. 1991 Mathematics Subject Classification. Primary, 58B34, 46L65 ; Secondary, 46L87 , 83C65 .