Dirac Triples for Unital AF Algebras
For a unital AF algebra A, we construct a family of triples (A, H, D) where A is represented faithfully on the Hilbert space H and D is an unbounded self-adjoint operator on H. These triples have the same properties as spectral triples except for the compact resolvent condition, so we call them Dirac triples. They serve as a generalization of Pearson-Bellissard spectral triples for an ultrametric Cantor set corresponding to choice functions. Pearson and Bellissard showed that the underlying ultrametric can be recovered by considering spectral triples associated to all choice functions. We obtain an analogue for unital AF algebras: the supremum of the Connes spectral distances induced by a large family of Dirac triples from our construction coincides with a generalized version of the Aguilar seminorm, which is a Leibniz Lip-norm for a unital AF algebra. Moreover, the convergence result of Aguilar is retained: equipped with the generalized Aguilar seminorm, a unital AF algebra is the limit of its defining finite-dimensional subalgebras for the quantum Gromov-Hausdorff propinquity.