Noncommutative Differential Geometry on Infinitesimal Spaces
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In this paper we use the language of noncommutative differential geometry to formalise discrete differential calculus. We begin with a brief review of inverse limit of posets as an approximation of topological spaces. We then show how to associate a C^*-algebra over a poset, giving it a piecewise-linear structure. Furthermore, we explain how dually the algebra of continuous function C(M) over a manifold M can be approximated by a direct limit of C^*-algebras over posets. Finally, in the spirit of noncommutative differential geometry, we define a finite dimensional spectral triple on each poset. We show how the usual finite difference calculus is recovered as the eigenvalues of the commutator with the Dirac operator. We prove a convergence result in the case of the d-lattice.
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