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 in $\mathbb{R}^d$ and for the torus $\mathbb{T}^d$.
翻译:在本文中,我们使用非交换微分几何的语言来形式化离散微分计算。我们首先简要回顾偏序集的逆极限作为拓扑空间的近似方法。然后,我们展示如何将$C^*$-代数与偏序集联系起来,并赋予其分段线性结构。此外,我们解释了$C^*$-代数在偏序集上的直极限如何与流形$M$上的连续函数代数$C(M)$相对应。最后,在非交换微分几何的思想下,我们在每个偏序集上定义了一个有限维谱三元组。我们展示了如何通过与Dirac算子的对易子的特征值来恢复通常的有限差分微积分。我们证明了在$d$-晶格上和环面$\mathbb{T}^d$上的情况下的收敛性结果。