2022/12/14 by Ryo Toyota, Toyota, Ryo
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Noncommutative and Quantum Gravity Theories
paper · pdf · doi:10.48550/arxiv.2212.07045
openalex publication_date 2022/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by the idea that our access to the spacetime is limited by the resolution of our measuring device, we give a new description of K-homology with a finite resolution. G. Yu introduced a C^*-algebra called the localization algebra C^*L(X) which consists of functions from [1,∞) to the Roe algebra C^*(X) whose propagations converge to 0 and he showed that for any finite dimensional simplicial complex X endowed with the spherical metric, the K-theory of the localization algebra is isomorphic to the K-homology of X. We give a coarse graining version of this theorem using controlled K-theory (also known as quantitative K-theory). Namely, instead of considering families of operators whose propagations converge to 0, we prove that for each dimension n, there exists a threshold rn>0 such that the K-homology of n-dimensional finite simplicial complex X is isomorphic to a certain group of equivalence classes of operators whose propagation is less than rn. This picture also enables us to represent any element in the K-homology group K_*(X) by a finite matrix for a finite simplicial complex X.